HVT: Predicting Cells with Layers using predictLayerHVT

Zubin Dowlaty, Srinivasan Sudarsanam, Somya Shambhawi

2023-10-26

1 Abstract

The HVT package is a collection of R functions to facilitate building topology preserving maps for rich multivariate data analysis. Tending towards a big data preponderance, a large number of rows. A collection of R functions for this typical workflow is organized below:

  1. Data Compression: Vector quantization (VQ), HVQ (hierarchical vector quantization) using means or medians. This step compresses the rows (long data frame) using a compression objective.

  2. Data Projection: Dimension projection of the compressed cells to 1D,2D or 3D with the Sammons Non-linear Algorithm. This step creates topology preserving map (also called as embedding) coordinates into the desired output dimension.

  3. Tessellation: Create cells required for object visualization using the Voronoi Tessellation method, package includes heatmap plots for hierarchical Voronoi tessellations (HVT). This step enables data insights, visualization, and interaction with the topology preserving map. Useful for semi-supervised tasks.

  4. Prediction: Scoring new data sets and recording their assignment using the map objects from the above steps, in a sequence of maps if required.

2 Example: HVT with the Personal Computer dataset

Data Understanding

In this vignette, we will use the Prices of Personal Computers dataset. This dataset contains 6259 observations and 6 features. The dataset observes the price from 1993 to 1995 of 486 personal computers in the US. The variables are price, speed, hd, ram, screen and ads.

Here, we load the data and store into a variable computers.

set.seed(240)
# Load data from csv files
computers <- read.csv("https://raw.githubusercontent.com/Mu-Sigma/HVT/master/vignettes/sample_dataset/Computers.csv")

Raw Personal Computers Dataset

The Computers dataset includes the following columns:

Let’s explore the Personal Computers Dataset containing (6259 points). For the shake of brevity we are displaying first six rows.

Table(head(computers), scroll = T, limit = 20)
X price speed hd ram screen cd multi premium ads trend
1 1499 25 80 4 14 no no yes 94 1
2 1795 33 85 2 14 no no yes 94 1
3 1595 25 170 4 15 no no yes 94 1
4 1849 25 170 8 14 no no no 94 1
5 3295 33 340 16 14 no no yes 94 1
6 3695 66 340 16 14 no no yes 94 1

Now, let us check the structure of the data and analyse its summary.

str(computers)
#> 'data.frame':    6259 obs. of  11 variables:
#>  $ X      : int  1 2 3 4 5 6 7 8 9 10 ...
#>  $ price  : int  1499 1795 1595 1849 3295 3695 1720 1995 2225 2575 ...
#>  $ speed  : int  25 33 25 25 33 66 25 50 50 50 ...
#>  $ hd     : int  80 85 170 170 340 340 170 85 210 210 ...
#>  $ ram    : int  4 2 4 8 16 16 4 2 8 4 ...
#>  $ screen : int  14 14 15 14 14 14 14 14 14 15 ...
#>  $ cd     : chr  "no" "no" "no" "no" ...
#>  $ multi  : chr  "no" "no" "no" "no" ...
#>  $ premium: chr  "yes" "yes" "yes" "no" ...
#>  $ ads    : int  94 94 94 94 94 94 94 94 94 94 ...
#>  $ trend  : int  1 1 1 1 1 1 1 1 1 1 ...
summary(computers)
#>        X            price          speed              hd              ram        
#>  Min.   :   1   Min.   : 949   Min.   : 25.00   Min.   :  80.0   Min.   : 2.000  
#>  1st Qu.:1566   1st Qu.:1794   1st Qu.: 33.00   1st Qu.: 214.0   1st Qu.: 4.000  
#>  Median :3130   Median :2144   Median : 50.00   Median : 340.0   Median : 8.000  
#>  Mean   :3130   Mean   :2220   Mean   : 52.01   Mean   : 416.6   Mean   : 8.287  
#>  3rd Qu.:4694   3rd Qu.:2595   3rd Qu.: 66.00   3rd Qu.: 528.0   3rd Qu.: 8.000  
#>  Max.   :6259   Max.   :5399   Max.   :100.00   Max.   :2100.0   Max.   :32.000  
#>      screen           cd               multi             premium               ads       
#>  Min.   :14.00   Length:6259        Length:6259        Length:6259        Min.   : 39.0  
#>  1st Qu.:14.00   Class :character   Class :character   Class :character   1st Qu.:162.5  
#>  Median :14.00   Mode  :character   Mode  :character   Mode  :character   Median :246.0  
#>  Mean   :14.61                                                            Mean   :221.3  
#>  3rd Qu.:15.00                                                            3rd Qu.:275.0  
#>  Max.   :17.00                                                            Max.   :339.0  
#>      trend      
#>  Min.   : 1.00  
#>  1st Qu.:10.00  
#>  Median :16.00  
#>  Mean   :15.93  
#>  3rd Qu.:21.50  
#>  Max.   :35.00

Let us first split the data into train and test. We will randomly select 80% of the data for training and remaining as testing.


num_rows <- nrow(computers)
set.seed(123)
train_indices <- sample(1:num_rows, 0.8 * num_rows)
trainComputers <- computers[train_indices, ]
testComputers <- computers[-train_indices, ]

K-means is not suitable for factor variables as the sample space for factor variables is discrete. A Euclidean distance function on such a space isn’t really meaningful. Hence, we will delete the factor variables(X, cd, multi, premium, trend) in our dataset.

trainComputers <-
  trainComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))
testComputers <-
  testComputers %>% dplyr::select(-c(X, cd, multi, premium, trend))

Raw Training Dataset

Now, lets have a look at the randomly selected raw training dataset containing (5007 data points). For the sake of brevity we are displaying first six rows.

trainComputers_data <- trainComputers %>% as.data.frame() %>% round(4)
trainComputers_data$Row.No <- as.numeric(row.names(trainComputers_data))
trainComputers_data <- trainComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
row.names(trainComputers_data) <- NULL
Table(head(trainComputers_data))
Row.No price speed hd ram screen ads
2463 2799 50 230 8 15 216
2511 2197 33 270 4 14 216
2227 2744 50 340 8 17 275
526 2999 66 245 16 15 139
4291 1974 33 200 4 14 248
2986 2490 33 528 16 14 267

Raw Testing Dataset

Now, lets have a look at the randomly selected raw testing dataset containing (1252 data points). For the sake of brevity we are displaying first six rows.

#testComputers <- scale(testComputers, center = scale_attr$`scaled:center`, scale = scale_attr$`scaled:scale`) 
testComputers_data <- testComputers %>% as.data.frame() %>% round(4)
testComputers_data$Row.No <- as.numeric(row.names(testComputers_data))
testComputers_data <- testComputers_data %>% dplyr::select(Row.No,price,speed,hd,ram,screen,ads)
rownames(testComputers_data) <- NULL
Table(head(testComputers_data))
Row.No price speed hd ram screen ads
3 1595 25 170 4 15 94
4 1849 25 170 8 14 94
7 1720 25 170 4 14 94
10 2575 50 210 4 15 94
11 2195 33 170 8 15 94
14 2295 25 245 8 14 94

3 Map A : Base Compressed Map

Let us try to visualize the compressed Map A from the flow diagram below.

Figure 1: Flow map with highlighted bounding box in red around compressed map A

Figure 1: Flow map with highlighted bounding box in red around compressed map A

This package can perform vector quantization using the following algorithms -

For more information on vector quantization, refer the following link.

The HVT function constructs highly compressed hierarchical Voronoi tessellations. The raw data is first scaled and this scaled data is supplied as input to the vector quantization algorithm. The vector quantization algorithm compresses the dataset until a user-defined compression percentage/rate is achieved using a parameter called quantization error which acts as a threshold and determines the compression percentage. It means that for a given user-defined compression percentage we get the ‘n’ number of cells, then all of these cells formed will have a quantization error less than the threshold quantization error.

Let’s try to comprehend the HVT function first before moving ahead.

HVT(
  dataset,
  min_compression_perc,
  n_cells,
  depth,
  quant.err,
  distance_metric = c("L1_Norm", "L2_Norm"),
  error_metric = c("mean", "max"),
  quant_method = c("kmeans", "kmedoids"),
  normalize = TRUE,
  diagnose = FALSE,
  hvt_validation = FALSE,
  train_validation_split_ratio = 0.8
)

Each of the parameters of HVT function have been explained below:

The output of HVT function (list of 6 elements) have been explained below:

We will use the HVT function to compress our data while preserving essential features of the dataset. Our goal is to achieve data compression upto atleast 80%. In situations where the compression ratio does not meet the desired target, we can explore adjusting the model parameters as a potential solution. This involves making modifications to parameters such as the quantization error threshold or increasing the number of cells and then rerunning the HVT function again.

In our example we will iteratively increase the number of cells until the desired compression percentage is reached instead of increasing the quantization threshold because it may reduce the level of detail captured in the data representation

First, we will construct map A by using the below mentioned model parameters.

3.0.1 Iteration 1:

We will pass the below mentioned model parameters along with training dataset (containing 5007 data points) to HVT function.

Model Parameters

set.seed(240)
map_A <- list()
map_A  <- HVT::HVT(
  trainComputers,
  n_cells = 200,
  depth = 1,
  quant.err = 0.2,
  projection.scale = 10,
  normalize = T,
  distance_metric = "L1_Norm",
  error_metric = "max",
  quant_method = "kmeans"
)

Let’s checkout the compression summary with n_cells set to 200.

compressionSummaryTable(map_A[[3]]$compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 200 79 0.4 n_cells: 200 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 42% of cells have reached the quantization threshold error. Therefore we can further subdivide the cells by increasing the n_cells parameters and then see if desired compression (80%) is reached

3.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression. Let’s try to compress again using the below mentioned set of model parameters and the Computers training dataset (containing 5007 data points) .

Model Parameters

map_A <- list()
map_A <-HVT::HVT(trainComputers,
                n_cells = 440,
                quant.err = 0.2,
                depth = 1,
                distance_metric = "L1_Norm",
                error_metric = "max",
                quant_method = "kmeans",
                normalize = T)

As per the manual, map_A[[3]] gives us detailed information about the hierarchical vector quantized data. map_A[[3]][['summary']] gives a nice tabular data containing no of points, Quantization Error and the codebook.

The datatable displayed below is the summary from map A showing Cell.ID, Centroids and Quantization Error for each of the 440 cells.

summaryTable(map_A[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 7 46 0.08 -0.76 -0.89 -0.88 -0.76 -0.67 1.57
1 1 2 10 108 0.08 -0.80 -0.89 -0.16 -0.76 -0.67 0.67
1 1 3 15 223 0.12 0.37 -0.89 -0.72 -0.05 0.43 -1.65
1 1 4 11 54 0.07 -1.50 -0.89 -0.75 -0.76 -0.67 0.62
1 1 5 8 146 0.13 -0.31 0.68 -0.95 -0.89 -0.67 -0.14
1 1 6 11 150 0.16 -0.66 0.68 -0.78 -0.79 -0.67 -0.73
1 1 7 11 170 0.1 0.03 -1.24 -0.13 -0.05 -0.67 0.38
1 1 8 8 334 0.15 0.62 2.30 0.08 -0.05 0.43 0.04
1 1 9 8 114 0.07 -0.16 0.68 -1.19 -1.11 -0.67 0.87
1 1 10 7 248 0.17 0.51 -0.08 0.34 -0.05 -0.67 -0.30
1 1 11 9 139 0.12 -0.01 0.68 -1.15 -1.00 -0.67 0.33
1 1 12 7 219 0.14 -1.36 0.24 0.46 -0.05 -0.67 -0.74
1 1 13 9 271 0.05 -1.08 0.68 0.49 -0.05 0.43 -0.84
1 1 14 19 109 0.06 -0.31 -0.89 -0.74 -0.76 -0.67 0.38
1 1 15 6 176 0.08 -0.72 -0.89 -0.07 -0.05 0.43 0.72
1 1 16 17 332 0.14 0.42 2.30 0.10 -0.05 0.43 1.50
1 1 17 12 18 0.05 -1.21 -1.27 -1.19 -1.11 -0.67 0.97
1 1 18 19 149 0.16 -0.68 -0.08 -0.46 -0.76 0.43 0.79
1 1 19 17 428 0.35 0.18 2.30 2.53 1.37 0.43 -2.22
1 1 20 20 320 0.36 0.82 -0.16 -0.09 -0.12 2.64 0.71
1 1 21 3 305 0.18 2.27 -0.35 -0.01 -0.05 -0.67 -1.32
1 1 22 7 227 0.1 -0.51 -0.89 0.45 -0.05 0.43 -0.47
1 1 23 10 178 0.12 0.00 0.68 -0.86 -0.76 -0.67 -0.90
1 1 24 9 365 0.1 0.68 -0.08 1.20 1.37 0.43 -0.36
1 1 25 5 14 0.11 -1.99 -0.89 -0.96 -1.11 -0.67 0.18
1 1 26 3 411 0.05 1.25 -0.89 2.29 2.79 0.43 0.57
1 1 27 18 122 0.15 -0.18 -0.98 -0.85 -0.76 0.43 0.68
1 1 28 15 189 0.11 0.40 -0.92 0.03 -0.05 -0.67 0.87
1 1 29 11 107 0.11 -0.49 -0.96 -0.88 -0.76 -0.67 -0.64
1 1 30 7 423 0.47 3.55 0.12 2.51 1.37 -0.67 0.44
1 1 31 14 90 0.05 -0.63 -0.89 -0.79 -0.76 -0.67 0.58
1 1 32 22 430 0.24 0.63 0.75 3.07 2.79 0.43 -2.27
1 1 33 5 390 0.3 1.37 -0.89 3.73 -0.19 -0.45 0.70
1 1 34 25 101 0.18 -0.85 -0.97 -0.71 -0.76 0.43 0.84
1 1 35 11 425 0.07 0.15 2.30 1.70 1.37 0.43 -2.39
1 1 36 10 358 0.05 0.24 -0.89 1.20 1.37 0.43 -0.84
1 1 37 16 166 0.11 0.03 0.68 -1.08 -0.78 -0.67 -1.65
1 1 38 13 45 0.05 -0.91 -0.89 -1.19 -1.11 -0.67 0.42
1 1 39 8 383 0.12 1.15 2.30 0.45 1.37 -0.67 -0.16
1 1 40 5 9 0.07 -1.24 -0.97 -1.19 -1.11 -0.67 1.57
1 1 41 11 419 0.06 1.41 -0.89 2.29 2.79 0.43 -0.82
1 1 42 8 242 0.14 -0.81 -0.08 0.30 -0.05 0.43 -0.65
1 1 43 13 179 0.09 0.41 0.68 -0.76 -0.76 -0.67 0.40
1 1 44 5 375 0.04 0.06 -0.08 1.70 1.37 0.43 -0.79
1 1 45 20 129 0.14 -1.15 0.68 -0.79 -0.76 -0.67 -0.40
1 1 46 10 292 0.22 0.86 0.68 -0.65 -0.12 0.43 -1.41
1 1 47 5 79 0.12 -0.89 -1.04 -0.94 -0.05 -0.67 1.02
1 1 48 23 246 0.11 -0.42 0.68 0.46 -0.05 -0.67 -0.63
1 1 49 8 207 0.25 0.74 -0.89 -0.40 -0.40 0.43 0.52
1 1 50 11 27 0.06 -1.06 -1.27 -1.19 -1.11 -0.67 0.43
1 1 51 8 51 0.11 -1.23 -0.08 -1.05 -0.85 -0.67 0.94
1 1 52 19 288 0.09 0.81 -0.89 0.45 1.37 -0.67 0.88
1 1 53 7 154 0.1 -0.62 0.68 -0.15 -0.76 -0.67 0.84
1 1 54 10 261 0.15 0.61 -0.08 -0.67 -0.05 0.43 -1.40
1 1 55 10 195 0.15 0.18 0.77 -0.09 -0.76 -0.67 0.83
1 1 56 14 250 0.09 0.52 0.68 -0.69 -0.05 -0.67 -1.61
1 1 57 20 331 0.2 -0.63 0.68 0.30 -0.76 2.64 -0.95
1 1 58 9 379 0.15 1.33 -0.08 -0.65 -0.29 2.64 -1.52
1 1 59 14 11 0.21 -0.28 -1.05 -0.79 -0.76 2.64 0.53
1 1 60 29 359 0.13 1.36 0.68 0.18 1.37 0.43 0.77
1 1 61 6 337 0.1 2.46 0.68 0.21 -0.05 -0.67 -0.87
1 1 62 6 1 0.17 -0.17 -1.21 -1.02 -0.76 2.64 1.32
1 1 63 28 243 0.33 -0.33 2.30 -0.23 -0.46 -0.67 -0.88
1 1 64 8 274 0.07 0.41 -1.27 0.45 1.37 -0.67 0.68
1 1 65 13 362 0.14 1.07 0.75 0.35 1.37 0.43 1.34
1 1 66 10 142 0.07 -0.34 -0.89 -0.80 -0.05 -0.67 -1.66
1 1 67 4 265 0.05 -0.55 0.68 1.23 -0.05 -0.67 -0.69
1 1 68 11 13 0.15 -0.83 -0.89 -0.25 -0.76 2.64 -0.33
1 1 69 8 298 0.17 -0.62 0.20 2.29 -0.05 -0.67 -0.95
1 1 70 4 335 0.06 1.34 -0.08 0.45 1.37 -0.67 -0.08
1 1 71 20 204 0.16 0.09 -0.08 0.02 -0.05 -0.67 0.86
1 1 72 10 43 0.06 -1.49 -0.89 -0.75 -0.76 -0.67 1.04
1 1 73 1 429 0 3.08 0.68 0.04 4.20 0.43 0.71
1 1 74 14 186 0.14 -0.79 -0.89 0.45 -0.05 -0.67 -0.68
1 1 75 4 410 0.37 2.27 0.68 3.73 -0.23 -0.40 0.68
1 1 76 9 163 0.16 1.05 -0.89 -0.41 -0.60 -0.67 0.61
1 1 77 10 400 0.07 -0.03 0.68 1.70 1.37 0.43 -2.38
1 1 78 6 275 0.18 1.14 0.68 0.13 -0.05 -0.67 -0.18
1 1 79 25 241 0.16 -0.88 0.68 0.40 -0.05 -0.67 -1.06
1 1 80 6 245 0.14 -1.22 0.68 -0.30 -0.05 0.43 -0.91
1 1 81 21 120 0.16 -0.46 -0.08 -0.82 -0.62 -0.67 0.70
1 1 82 11 40 0.18 -0.93 -0.99 -1.19 -1.11 0.43 0.37
1 1 83 9 342 0.28 1.16 0.43 -0.52 -0.68 2.64 1.12
1 1 84 8 286 0.05 -0.72 1.11 0.50 -0.05 0.43 -0.83
1 1 85 8 33 0.1 -1.06 -0.89 -1.18 -1.02 -0.67 -0.99
1 1 86 5 282 0.23 -1.50 0.53 0.19 -0.48 0.43 -2.16
1 1 87 17 137 0.07 -0.31 0.68 -0.78 -0.76 -0.67 0.96
1 1 88 16 169 0.05 -0.03 -0.89 0.05 -0.05 -0.67 1.02
1 1 89 7 290 0.05 1.10 -0.89 0.15 1.37 -0.67 0.36
1 1 90 6 24 0.19 -0.97 -1.02 -1.07 -0.94 0.43 -1.40
1 1 91 4 434 0.59 4.07 1.49 1.29 0.30 2.64 0.07
1 1 92 19 409 0.65 0.92 0.42 1.41 1.37 2.64 -0.60
1 1 93 9 393 0.46 2.04 2.30 0.91 -0.05 -0.31 -0.34
1 1 94 8 22 0.08 -1.66 -1.27 -0.84 -0.76 -0.67 0.84
1 1 95 23 158 0.22 -1.41 0.68 -0.20 -0.70 -0.67 -1.05
1 1 96 11 330 0.22 1.55 0.68 -0.45 -0.31 0.43 -1.59
1 1 97 6 144 0.17 0.49 -0.89 -0.64 -0.76 -0.67 -0.17
1 1 98 12 121 0.15 0.21 -0.89 -0.80 -0.76 -0.67 -1.70
1 1 99 14 329 0.24 2.06 0.63 0.31 -0.25 0.43 0.99
1 1 100 16 299 0.05 1.21 -0.89 0.46 1.37 -0.67 0.86
1 1 101 5 328 0.1 -0.87 1.11 1.33 -0.05 0.43 -1.13
1 1 102 10 278 0.1 0.33 -0.93 0.45 1.37 -0.67 0.02
1 1 103 5 102 0.1 -1.05 0.68 -0.76 -0.76 -0.67 1.27
1 1 104 12 25 0.08 -1.25 -1.05 -0.78 -0.76 -0.67 1.57
1 1 105 9 385 0.25 2.05 0.34 -0.21 -0.05 2.64 1.12
1 1 106 10 231 0.18 -0.02 -0.08 0.02 -0.05 0.43 1.29
1 1 107 5 193 0.14 -0.66 -0.08 -0.33 -0.05 -0.67 -1.01
1 1 108 4 418 0.06 -0.03 0.68 3.07 1.37 0.43 -2.25
1 1 109 4 306 0.16 1.83 0.68 -0.59 -0.05 -0.67 -1.57
1 1 110 5 378 0.15 1.09 0.68 1.20 1.37 0.43 -0.11
1 1 111 2 308 0.07 -0.55 0.68 0.26 1.37 -0.67 -1.25
1 1 112 18 239 0.15 0.44 0.75 0.08 -0.05 -0.67 1.05
1 1 113 11 62 0.06 -1.14 -0.89 -0.81 -0.76 -0.67 0.73
1 1 114 5 168 0.07 -0.26 -0.89 0.47 -0.05 -0.67 1.40
1 1 115 18 415 0.11 0.74 -0.08 2.29 2.79 0.43 -0.93
1 1 116 5 97 0.13 0.23 -1.19 -1.05 -0.76 -0.67 0.77
1 1 117 24 19 0.22 -0.06 2.30 -0.89 -0.89 -0.67 1.18
1 1 118 9 184 0.17 -0.98 0.34 -0.29 -0.05 -0.67 -0.08
1 1 119 11 42 0.12 -0.82 -0.89 -1.04 -0.79 -0.67 -1.65
1 1 120 9 257 0.29 2.17 -0.62 0.10 -0.05 -0.55 0.65
1 1 121 27 155 0.18 0.15 0.68 -0.84 -0.76 -0.67 0.81
1 1 122 20 215 0.11 0.33 -0.08 -0.69 -0.05 -0.67 -1.64
1 1 123 14 348 0.24 0.61 0.74 0.37 1.37 0.43 0.25
1 1 124 13 162 0.12 0.01 -1.01 -0.67 -0.05 -0.67 -1.57
1 1 125 7 366 0.22 1.62 0.35 -0.29 1.37 -0.67 -1.65
1 1 126 9 77 0.02 -0.88 -0.89 -0.78 -0.76 -0.67 0.66
1 1 127 25 253 0.14 0.70 0.68 0.16 -0.05 -0.67 0.61
1 1 128 9 84 0.09 -0.27 -1.27 -0.81 -0.76 -0.67 0.81
1 1 129 3 398 0.04 -0.03 -0.08 2.29 1.37 0.43 -1.98
1 1 130 9 412 0.05 0.90 -0.89 2.29 2.79 0.43 -0.48
1 1 131 8 309 0.06 1.43 -0.89 0.41 1.37 -0.67 0.46
1 1 132 8 267 0.08 0.10 0.68 0.37 -0.05 0.43 1.57
1 1 133 16 252 0.22 0.94 -0.08 -0.40 -0.14 0.43 0.43
1 1 134 15 119 0.13 -0.60 -0.99 -0.75 -0.76 0.43 0.24
1 1 135 13 48 0.09 -0.66 -0.92 -1.19 -1.11 -0.67 0.82
1 1 136 9 326 0.11 1.45 -0.08 0.32 1.37 -0.67 0.38
1 1 137 11 386 0.22 1.37 0.68 -0.66 -0.18 2.64 -1.36
1 1 138 20 255 0.15 0.43 -0.89 -0.34 -0.05 2.64 0.68
1 1 139 8 161 0.14 -0.68 -0.94 -0.08 -0.05 0.43 1.41
1 1 140 10 38 0.13 -1.22 -1.19 -0.92 -0.76 0.43 0.91
1 1 141 14 180 0.12 0.09 0.68 -0.56 -0.76 -0.67 0.07
1 1 142 13 427 0.21 2.01 0.62 2.29 2.79 0.43 -0.18
1 1 143 15 354 0.13 1.63 0.68 0.39 1.37 -0.67 0.26
1 1 144 11 301 0.14 0.82 -1.03 0.15 1.37 -0.67 -0.85
1 1 145 7 205 0.13 0.24 -0.08 -0.05 -0.05 -0.67 1.57
1 1 146 10 327 0.14 0.66 0.68 0.60 1.37 -0.67 0.42
1 1 147 11 403 0.11 0.92 2.30 1.22 1.37 0.43 -0.79
1 1 148 7 127 0.08 -0.20 -0.89 -0.10 -0.76 -0.67 0.69
1 1 149 6 217 0.11 0.41 -0.89 0.26 -0.05 -0.67 -0.34
1 1 150 11 323 0.1 0.96 -0.89 0.46 1.37 0.43 0.72
1 1 151 9 208 0.16 0.16 0.68 -0.60 -0.05 -0.67 0.43
1 1 152 14 405 0.14 -0.04 0.68 2.29 1.37 0.43 -2.24
1 1 153 10 98 0.08 -0.22 -0.89 -1.14 -0.76 -0.67 0.33
1 1 154 7 47 0.05 -0.99 -0.89 -1.16 -0.76 -0.67 1.03
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1 1 412 6 426 0.2 3.36 0.68 -0.27 1.37 2.64 -0.22
1 1 413 12 285 0.1 0.72 -1.24 0.30 1.37 -0.67 0.32
1 1 414 8 363 0.08 1.28 1.11 0.45 1.37 -0.67 1.36
1 1 415 8 437 0.27 1.02 1.00 3.07 2.79 2.64 -2.30
1 1 416 7 284 0.19 0.20 -0.66 -0.74 -0.05 2.64 -0.71
1 1 417 9 264 0.06 0.29 -0.89 0.45 1.37 -0.67 0.66
1 1 418 21 238 0.21 0.24 -0.08 -0.09 -0.05 0.43 0.69
1 1 419 3 296 0.02 -0.47 0.68 0.03 -0.05 0.43 -2.34
1 1 420 10 100 0.16 -0.44 -0.97 -0.87 -0.76 0.43 1.21
1 1 421 11 174 0.15 -0.61 0.68 -0.02 -0.76 -0.67 0.17
1 1 422 9 311 0.25 1.52 -0.53 -0.55 -0.76 2.64 0.36
1 1 423 9 145 0.12 -0.52 -1.14 0.09 -0.05 -0.67 1.24
1 1 424 5 368 0.15 2.37 0.68 0.36 1.37 -0.67 0.68
1 1 425 9 373 0.12 1.65 0.68 0.39 1.37 0.43 -0.01
1 1 426 10 228 0.13 0.42 -0.08 -0.33 -0.05 -0.67 -0.92
1 1 427 20 86 0.22 -1.23 -0.93 -0.85 -0.83 0.43 -0.46
1 1 428 7 106 0.06 -0.07 -0.89 -0.76 -0.76 -0.67 0.98
1 1 429 8 135 0.12 -0.45 -0.99 -0.78 -0.76 0.43 -0.77
1 1 430 8 30 0.12 -0.51 1.05 -1.19 -1.11 -0.67 1.36
1 1 431 19 357 0.16 1.03 0.68 -0.27 -0.05 2.64 0.61
1 1 432 20 406 0.38 0.02 2.30 0.99 -0.05 2.64 -1.12
1 1 433 10 302 0.26 1.83 0.53 -0.09 -0.33 0.43 0.24
1 1 434 5 73 0.04 -0.72 -0.89 -0.95 -0.76 -0.67 0.82
1 1 435 1 440 0 5.51 0.68 3.07 4.20 2.64 0.50
1 1 436 14 300 0.29 2.04 0.68 0.38 -0.15 -0.67 0.69
1 1 437 19 74 0.1 -1.10 -0.89 -0.81 -0.76 -0.67 -0.90
1 1 438 7 347 0.13 1.21 -0.08 0.41 1.37 0.43 0.59
1 1 439 31 374 0.28 -0.27 0.87 0.49 -0.05 2.64 -1.24
1 1 440 17 233 0.2 -0.48 0.68 -0.10 -0.13 0.43 0.60

Now let us understand what each column in the above table means:

All the columns after this will contain centroids for each cell. They can also be called a codebook, which represents a collection of all centroids or codewords.

Now, let’s check the compression summary for HVT (map A) where n_cell was set to 440. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapA_compression_summary <- map_A[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapA_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 440 355 0.81 n_cells: 440 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 81% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

Now let’s try to understand plotHVT function. The parameters have been explained in detail below:

plotHVT(hvt.results, line.width, color.vec, pch1 = 21, centroid.size = 3, title = NULL, maxDepth = 1)

Let’s plot the Voronoi tessellation for layer 1 (map A).

HVT::plotHVT(map_A,
        line.width = c(0.4), 
        color.vec = c("#141B41"),
        centroid.size = 0.01,
        maxDepth = 1) 
Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Figure 2: The Voronoi Tessellation for layer 1 (map A) shown for the 440 cells in the dataset ’computers’

Heat Maps

We will now overlay all the features as heatmap over the Voronoi Tessellation plot for better visualization and identification of patterns, trends, and variations in the data.

Let’s have a look at the function hvtHmap that we will use to overlay features as heatmap.

hvtHmap(hvt.results, dataset, child.level, hmap.cols, color.vec ,line.width, palette.color = 6)

Now let’s plot the Voronoi Tessellation with the heatmap overlaid for all the features in the torus data for better visualization and interpretation of data patterns and distributions.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "n",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell

Figure 3: The Voronoi Tessellation with the heat map overlaid for No. of entities in each cell


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "price",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset

Figure 4: The Voronoi Tessellation with the heat map overlaid for variable ’price’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "speed",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset

Figure 5: The Voronoi Tessellation with the heat map overlaid for variable ’speed’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "hd",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset

Figure 6: The Voronoi Tessellation with the heat map overlaid for variable ’hd’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ram",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset

Figure 7: The Voronoi Tessellation with the heat map overlaid for variable ’ram’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "screen",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset

Figure 8: The Voronoi Tessellation with the heat map overlaid for variable ’screen’ in the ’computers’ dataset


  hvtHmap(
  map_A,
  trainComputers,
  child.level = 1,
  hmap.cols = "ads",
  line.width = c(0.2),
  color.vec = c("#141B41"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 440,
) 
Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

Figure 9: The Voronoi Tessellation with the heat map overlaid for variable ’ads’ in the ’computers’ dataset

4 Map B : Compressed Novelty Map

Let us try to visualize the Map B from the flow diagram below.

Figure 10: Flow map with highlighted bounding box in red around map B

Figure 10: Flow map with highlighted bounding box in red around map B

In this section, we will manually figure out the novelty cells from the plotted map A and store it in identified_Novelty_cells variable.

Note: For manual selecting the novelty cells from map A, one can enhance its interactivity by adding plotly elements to the code. This will transform map A into an interactive plot, allowing users to actively engage with the data. By hovering over the centroids of the cells, a tag containing segment child information will be displayed. Users can explore the map by hovering over different cells and selectively choose the novelty cells they wish to consider. Added an image for reference.

Figure 11: Manually selecting novelty cells

Figure 11: Manually selecting novelty cells

The removeNovelty function removes the identified novelty cell(s) from the training dataset (containing 5007 datapoints) and stores those records separately.

It takes input as the cell number (Segment.Child) of the manually identified novelty cell(s) and the compressed HVT map (map A) with 440 cells. It returns a list of two items: dataset with novelty records, and a subset of the dataset without the novelty records.

identified_Novelty_cells <<- c(73,321,332,338,435)   #73,321,332,338,435
output_list <- removeNovelty(identified_Novelty_cells, map_A)

[1] “The following cell(s) have been removed as novelties from the dataset: 73 321 332 338 435”

data_with_novelty <- output_list[[1]]
dataset_without_novelty <- output_list[[2]]

Let’s have a look at the data with novelties(containing 24 records). For the sake of brevity, we will only show the first 20 rows.

novelty_data <- data_with_novelty
novelty_data$Row.No <- row.names(novelty_data)
novelty_data <- novelty_data %>% dplyr::select("Row.No","Cell.ID","Cell.Number","price","speed","hd","ram","screen","ads")
colnames(novelty_data) <- c("Row.No","Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
novelty_data %>% head(100) %>% 
  as.data.frame() %>%
  Table(scroll = T, limit = 20)
Row.No Cell.ID Segment.Child price speed hd ram screen ads
1 429 73 3.0762240 0.6794579 0.0421969 4.2031619 0.4307274 0.7120258
2 438 321 2.6449847 0.6794579 6.5710368 1.3676416 0.4307274 0.6851382
3 438 321 1.6578103 2.2969425 6.5710368 1.3676416 0.4307274 -1.2507722
4 439 332 0.9130997 0.6794579 4.6232922 2.7854017 0.4307274 -2.4607161
5 439 332 1.8569770 1.1076156 4.6232922 4.2031619 2.6404120 -2.4607161
6 439 332 1.5192595 2.2969425 4.6232922 4.2031619 0.4307274 -2.4607161
7 439 332 1.4482522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
8 439 332 1.8569770 1.1076156 4.6232922 4.2031619 2.6404120 -2.4607161
9 439 332 1.1555636 2.2969425 4.6232922 2.7854017 0.4307274 -2.4607161
10 439 332 1.7773103 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
11 439 332 1.3460710 0.6794579 4.6232922 4.2031619 0.4307274 -2.4607161
12 439 332 1.4482522 1.1076156 4.6232922 4.2031619 0.4307274 -2.4607161
13 218 338 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903
14 218 338 -1.5115391 0.6794579 0.4473277 -0.7589986 -0.6741149 -1.9767386
15 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465
16 218 338 -1.4959522 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723
17 218 338 -1.5981334 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2859465
18 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.2993903
19 218 338 -1.1668940 0.6794579 0.4473277 -0.7589986 -0.6741149 -2.4472723
20 218 338 -1.3227637 0.6794579 0.4473277 -0.7589986 -0.6741149 -1.9767386

4.1 Voronoi Tessellation with highlighted novelty cell

The plotCells function is used to plot the Voronoi tessellation using the compressed HVT map (map A) containing 440 cells and highlights the identified novelty cell(s) i.e 5 cells (containing 24 records) in red on the map.

plotCells(identified_Novelty_cells, map_A,line.width = c(0.4),centroid.size = 0.01)
Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

Figure 12: The Voronoi Tessellation constructed using the compressed HVT map (map A) with the novelty cell(s) highlighted in red

We pass the dataframe with novelty records (24 records) to HVT function along with other model parameters mentioned below to generate map B (layer2)

Model Parameters

colnames(data_with_novelty) <- c("Cell.ID","Segment.Child","price","speed","hd","ram","screen","ads")
dataset_with_novelty <- data_with_novelty[,-1:-2]
map_B <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_B <- HVT::HVT(dataset_with_novelty,
                  n_cells = 12 ,   #6
                  depth = 1,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F
                  )

The datatable displayed below is the summary from map B (layer 2) showing Cell.ID, Centroids and Quantization Error for each of the 12 cells.

summaryTable(map_B[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 1 6 0 -1.08 0.68 0.45 -0.76 -0.67 -1.98
1 1 2 1 3 0 -1.51 0.68 0.45 -0.76 -0.67 -1.98
1 1 3 2 1 0.01 -1.55 -0.08 0.45 -0.76 -0.67 -1.98
1 1 4 2 11 0 1.86 1.11 4.62 4.20 2.64 -2.46
1 1 5 1 7 0 3.08 0.68 0.04 4.20 0.43 0.71
1 1 6 5 8 0.31 1.39 0.94 4.62 3.92 0.43 -2.46
1 1 7 1 12 0 5.51 0.68 3.07 4.20 2.64 0.50
1 1 8 2 9 0.15 1.34 2.30 4.62 3.49 0.43 -2.46
1 1 9 1 4 0 -1.32 0.68 0.45 -0.76 -0.67 -1.98
1 1 10 3 2 0.02 -1.53 0.68 0.45 -0.76 -0.67 -2.34
1 1 11 2 10 0.38 2.15 1.49 6.57 1.37 0.43 -0.28
1 1 12 3 5 0.02 -1.17 0.68 0.45 -0.76 -0.67 -2.34

Now let’s check the compression summary for HVT (map B). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapB_compression_summary <- map_B[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapB_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 12 10 0.83 n_cells: 12 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 83% of the cells have hit the quantization threshold error.Since we are successfully able to attain the desired compression percentage, so we will not further subdivide the cells

5 Map C : Compressed Map without Novelty

Let us try to visualize the compressed Map C from the flow diagram below.

Figure 13:Flow map with highlighted bounding box in red around compressed map C

Figure 13:Flow map with highlighted bounding box in red around compressed map C

5.0.1 Iteration 1:

With the Novelties removed, we construct another hierarchical Voronoi tessellation map C layer 2 on the dataset without Novelty (containing 4983 records) and below mentioned model parameters.

Model Parameters

map_C <- list()
mapA_scale_summary = map_A[[3]]$scale_summary
map_C <- HVT::HVT(dataset_without_novelty,
                  n_cells = 10,
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

Now let’s check the compression summary for HVT (map C) where n_cell was set to 10. The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 10 0 0 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 100 7 0.07 n_cells: 10 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 7% of the cells have hit the quantization threshold error in level 2

5.0.2 Iteration 2:

Since, we are yet to achive atleast 80% compression at depth 2. Let’s try to compress again using the below mentioned set of model parameters and the dataset without novelty (containing 4983 records).

Model Parameters

map_C <- list()
map_C <- HVT::HVT(dataset_without_novelty,
                  n_cells = 23,    #23
                  depth = 2,
                  quant.err = 0.2,
                  projection.scale = 10,
                  normalize = F,
                  distance_metric = "L1_Norm",
                  error_metric = "max",
                  quant_method = "kmeans",
                  diagnose = F,
                  scale_summary = mapA_scale_summary)

The datatable displayed below is the summary from map C (layer2). showing Cell.ID, Centroids and Quantization Error for each of the 531 cells.

summaryTable(map_C[[3]]$summary,scroll = T,limit = 500)
Segment.Level Segment.Parent Segment.Child n Cell.ID Quant.Error price speed hd ram screen ads
1 1 1 391 512 0.56 -1.27 -1.04 -0.91 -0.83 -0.59 0.86
1 1 2 105 96 1.02 2.08 0.75 0.11 0.71 2.64 0.56
1 1 3 190 55 1.37 1.48 0.02 2.32 2.63 0.12 -0.22
1 1 4 86 15 0.94 0.87 1.27 2.61 2.57 1.33 -1.76
1 1 5 257 326 0.49 0.41 0.45 -0.08 -0.20 -0.67 0.66
1 1 6 116 201 0.77 1.57 0.45 -0.24 0.27 -0.30 -1.30
1 1 7 235 228 0.58 0.85 -0.82 0.42 1.37 -0.47 0.52
1 1 8 149 185 0.84 -0.79 1.73 0.76 -0.22 -0.28 -1.75
1 1 9 353 306 0.69 -0.77 0.55 0.28 -0.22 -0.17 -0.79
1 1 10 234 297 0.54 0.25 0.50 -0.11 -0.16 0.43 0.66
1 1 11 330 478 0.56 -1.18 -0.79 -0.70 -0.76 -0.52 -0.69
1 1 12 373 434 0.6 -0.50 0.40 -0.84 -0.84 -0.56 0.66
1 1 13 209 147 0.81 1.32 0.99 0.36 1.37 -0.01 0.70
1 1 14 337 471 0.36 -0.51 -0.97 -0.87 -0.75 -0.67 0.57
1 1 15 283 404 0.71 0.06 -0.40 -0.78 -0.40 -0.42 -1.42
1 1 16 168 142 0.85 0.13 0.66 0.48 -0.02 2.64 -0.97
1 1 17 285 384 0.53 -0.04 -0.96 0.05 -0.12 -0.65 0.54
1 1 18 118 301 0.75 0.35 2.30 -0.13 -0.29 -0.28 0.82
1 1 19 256 414 0.47 -0.47 -0.84 -0.43 -0.48 0.43 0.60
1 1 20 78 57 1 0.18 1.67 2.05 1.31 0.39 -1.97
1 1 21 171 360 0.64 0.24 -0.46 -0.37 -0.41 2.64 0.59
1 1 22 170 130 1.28 0.28 -0.01 1.64 0.99 0.21 -0.62
1 1 23 89 219 0.88 1.98 0.59 0.20 -0.20 0.12 0.58
2 1 1 19 528 0.09 -1.31 -1.17 -1.19 -1.11 -0.67 0.99
2 1 2 18 479 0.31 -1.42 -0.87 -0.21 -0.72 -0.67 0.53
2 1 3 28 521 0.1 -1.47 -1.22 -1.11 -0.76 -0.67 0.89
2 1 4 10 530 0.14 -1.84 -1.04 -1.20 -1.11 -0.67 0.79
2 1 5 27 519 0.1 -1.20 -1.09 -1.19 -1.11 -0.67 0.50
2 1 6 13 524 0.18 -1.36 -1.10 -0.99 -0.87 0.43 1.36
2 1 7 5 523 0.04 -1.19 -1.27 -0.78 -0.76 -0.67 1.57
2 1 8 23 494 0.06 -1.18 -0.89 -0.78 -0.76 -0.67 0.72
2 1 9 27 514 0.14 -1.35 -1.27 -1.08 -0.76 -0.67 0.39
2 1 10 35 510 0.1 -1.08 -1.27 -0.88 -0.76 -0.67 0.90
2 1 11 6 529 0.15 -1.22 -0.62 -1.19 -1.11 -0.67 1.50
2 1 12 10 513 0.07 -0.95 -0.89 -0.85 -0.76 -0.67 1.57
2 1 13 10 517 0.22 -1.70 -1.19 -0.73 -0.83 -0.67 0.57
2 1 14 11 472 0.17 -1.19 -1.13 -0.33 -0.05 -0.67 0.97
2 1 15 11 507 0.19 -1.00 -0.93 -1.17 -0.63 -0.67 0.97
2 1 16 7 531 0.07 -1.58 -1.27 -1.18 -0.96 -0.67 1.57
2 1 17 26 499 0.09 -1.31 -0.89 -0.92 -0.76 -0.67 0.36
2 1 18 23 515 0.07 -0.99 -0.89 -1.19 -1.11 -0.67 0.92
2 1 19 13 520 0.14 -1.56 -0.98 -0.74 -0.76 -0.67 1.34
2 1 20 15 489 0.12 -1.05 -0.89 -0.26 -0.76 -0.67 1.31
2 1 21 16 503 0.18 -1.48 -0.79 -0.73 -0.76 -0.67 0.76
2 1 22 15 518 0.26 -1.37 -1.07 -1.09 -1.07 0.43 0.66
2 1 23 23 500 0.07 -1.03 -0.89 -0.82 -0.76 -0.67 1.06
2 2 1 4 108 0.12 2.31 0.68 -0.12 -0.05 2.64 1.31
2 2 2 3 118 0.11 2.53 0.68 -0.47 -0.76 2.64 0.36
2 2 3 7 99 0.21 1.06 0.80 0.29 1.37 2.64 0.35
2 2 4 2 89 0.06 1.09 2.30 0.25 -0.05 2.64 1.57
2 2 5 7 59 0.13 3.21 0.68 -0.27 1.37 2.64 0.53
2 2 6 7 88 0.11 1.85 0.68 0.16 1.37 2.64 0.85
2 2 7 2 145 0.11 1.14 0.68 0.10 -0.05 2.64 1.36
2 2 8 3 38 0.11 3.32 0.68 -0.27 1.37 2.64 -0.78
2 2 9 6 100 0.17 2.86 0.68 0.10 -0.05 2.64 0.48
2 2 10 2 24 0.11 3.50 2.30 1.23 -0.05 2.64 -0.23
2 2 11 2 12 0.38 4.64 0.68 1.36 0.66 2.64 0.37
2 2 12 6 56 0.25 1.52 2.30 0.32 1.37 2.64 0.77
2 2 13 4 50 0.34 1.64 0.49 1.90 1.37 2.64 0.14
2 2 14 13 76 0.22 2.42 0.68 0.03 1.37 2.64 0.53
2 2 15 4 103 0.12 1.08 2.30 0.14 -0.05 2.64 0.52
2 2 16 11 167 0.13 1.23 0.68 -0.20 -0.05 2.64 0.59
2 2 17 1 107 0 1.99 -0.08 0.49 -0.05 2.64 -0.62
2 2 18 4 84 0.15 1.52 0.68 0.34 1.37 2.64 1.36
2 2 19 5 169 0.17 1.81 -0.08 -0.29 -0.19 2.64 0.99
2 2 20 4 70 0.11 2.69 -0.89 -0.27 1.37 2.64 0.69
2 2 21 4 122 0.11 1.99 0.68 0.22 -0.05 2.64 0.57
2 2 22 3 60 0.12 2.74 -0.89 -0.27 1.37 2.64 -0.78
2 2 23 1 195 0 1.63 0.68 -0.95 -0.76 2.64 0.69
2 3 1 5 39 0.04 1.54 -0.08 2.29 2.79 0.43 -0.81
2 3 2 3 9 0.05 1.71 -0.89 2.29 2.79 2.64 -0.55
2 3 3 10 51 0.07 1.10 -0.08 2.29 2.79 0.43 -0.56
2 3 4 5 32 0.39 3.74 -0.10 2.60 1.37 -0.67 0.96
2 3 5 3 19 0.02 2.16 2.30 2.29 2.79 -0.67 0.04
2 3 6 1 20 0 2.03 2.30 2.29 2.79 0.43 0.35
2 3 7 16 49 0.18 1.49 -0.89 2.29 2.79 0.43 -0.06
2 3 8 7 40 0.09 1.80 -0.08 2.29 2.79 0.43 -0.26
2 3 9 2 54 0.05 2.90 0.68 3.73 -0.05 -0.67 0.69
2 3 10 1 91 0 1.16 0.68 2.29 1.37 -0.67 0.04
2 3 11 7 46 0.05 1.16 0.68 2.29 2.79 0.43 -0.49
2 3 12 4 77 0.12 1.66 0.68 2.44 1.37 0.43 0.80
2 3 13 7 37 0.08 1.63 0.68 2.29 2.79 0.43 -0.73
2 3 14 12 53 0.1 1.69 0.68 2.29 2.79 -0.67 0.38
2 3 15 2 43 0.04 3.07 0.68 2.29 1.37 -0.67 -0.87
2 3 16 5 23 0.04 2.06 2.30 2.29 2.79 -0.67 0.35
2 3 17 12 45 0.06 0.92 0.68 2.29 2.79 0.43 -0.86
2 3 18 5 82 0.13 1.35 0.68 2.47 1.37 0.43 0.11
2 3 19 5 17 0.04 2.23 2.30 2.29 2.79 -0.67 0.69
2 3 20 24 61 0.12 1.30 -0.89 2.29 2.79 -0.67 0.33
2 3 21 9 36 0.18 2.07 0.68 2.29 2.79 0.43 -0.03
2 3 22 13 52 0.11 0.65 -0.08 2.29 2.79 0.43 -1.03
2 3 23 32 48 0.13 1.15 -0.89 2.29 2.79 0.43 -0.66
2 4 1 5 7 0.11 0.51 2.30 2.05 1.37 2.64 -2.33
2 4 2 8 18 0.07 0.52 0.68 3.07 2.79 0.43 -2.36
2 4 3 6 41 0.05 0.79 0.68 2.29 2.79 0.43 -1.24
2 4 4 2 3 0 1.08 2.30 3.07 2.79 0.43 -2.46
2 4 5 3 8 0.13 1.91 0.43 2.29 2.79 2.64 -0.55
2 4 6 1 11 0 0.01 1.11 3.07 2.79 0.43 -2.45
2 4 7 3 26 0.03 0.73 -0.08 3.07 2.79 0.43 -1.98
2 4 8 1 4 0 0.99 -0.08 3.07 2.79 2.64 -1.98
2 4 9 3 2 0.05 1.48 2.30 2.29 2.79 2.64 -1.08
2 4 10 2 14 0.02 0.81 0.68 3.07 2.79 0.43 -2.46
2 4 11 1 6 0 0.74 2.30 3.07 2.79 0.43 -1.98
2 4 12 2 16 0.02 0.89 1.11 3.07 2.79 0.43 -1.98
2 4 13 2 29 0.04 0.84 2.30 1.70 1.37 2.64 -1.02
2 4 14 6 5 0.03 0.74 2.30 3.07 2.79 0.43 -2.40
2 4 15 7 1 0.23 1.03 1.16 3.07 2.79 2.64 -2.34
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2 4 17 6 22 0.25 0.85 -0.22 2.29 2.79 2.64 -0.97
2 4 18 2 13 0.05 1.21 0.68 2.29 2.79 2.64 -1.10
2 4 19 5 30 0.11 0.32 0.68 2.17 1.37 2.64 -2.29
2 4 20 6 10 0.04 0.69 1.11 3.07 2.79 0.43 -2.32
2 4 21 1 34 0 0.30 -0.08 2.29 1.37 2.64 -1.98
2 4 22 6 21 0.06 1.46 2.30 2.29 2.79 0.43 -1.01
2 4 23 3 28 0.01 1.00 2.30 2.29 2.79 0.43 -0.79
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2 5 5 6 311 0.06 -0.28 0.68 0.45 -0.05 -0.67 0.30
2 5 6 8 287 0.15 0.45 0.68 0.16 -0.05 -0.67 -0.20
2 5 7 5 305 0.06 -0.07 0.68 0.45 -0.05 -0.67 0.48
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2 5 12 7 314 0.16 1.02 0.68 -0.39 -0.15 -0.67 1.12
2 5 13 17 363 0.2 0.66 0.63 -0.69 -0.76 -0.67 0.15
2 5 14 15 278 0.11 1.06 0.68 0.06 -0.05 -0.67 0.76
2 5 15 8 356 0.16 0.13 0.79 -0.13 -0.76 -0.67 0.38
2 5 16 11 275 0.19 1.06 0.68 -0.09 -0.05 -0.67 0.19
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2 5 20 7 386 0.14 1.08 -0.08 -0.62 -0.76 -0.67 0.63
2 5 21 12 336 0.2 -0.17 0.68 -0.04 -0.05 -0.67 0.86
2 5 22 7 364 0.13 0.24 -0.08 -0.05 -0.05 -0.67 1.57
2 5 23 0 NA NA NA NA NA NA NA NA
2 6 1 6 184 0.1 2.46 0.68 0.21 -0.05 -0.67 -0.87
2 6 2 9 133 0.29 1.43 0.51 -0.44 1.37 0.43 -1.15
2 6 3 2 233 0.08 1.08 0.68 0.47 -0.05 -0.67 -0.71
2 6 4 9 247 0.22 0.90 0.59 -0.69 -0.13 0.43 -1.39
2 6 5 2 194 0.08 2.30 0.30 -0.32 -0.05 -0.67 -1.71
2 6 6 8 238 0.26 1.04 0.58 -0.29 -0.14 0.43 -0.77
2 6 7 9 127 0.06 2.98 0.68 0.28 -0.05 -0.67 -1.67
2 6 8 4 165 0.11 1.26 -0.08 -0.07 1.37 -0.67 -1.37
2 6 9 5 124 0.19 1.76 0.53 -0.29 1.37 -0.67 -1.66
2 6 10 1 101 0 3.77 0.68 0.15 -0.05 -0.67 -1.72
2 6 11 5 292 0.22 1.44 0.53 -0.77 -0.33 -0.67 -1.67
2 6 12 1 365 0 1.35 -0.08 -0.67 -0.76 -0.67 -0.62
2 6 13 4 149 0.06 1.43 0.68 0.15 1.37 -0.67 -0.87
2 6 14 10 197 0.2 1.57 0.68 -0.47 -0.33 0.43 -1.64
2 6 15 3 159 0.2 1.48 2.30 0.28 -0.05 -0.67 -0.87
2 6 16 3 280 0.1 0.85 -0.08 0.22 -0.05 -0.67 -0.68
2 6 17 9 191 0.15 1.09 -0.93 -0.29 1.37 -0.67 -1.62
2 6 18 4 140 0.08 3.03 -0.08 0.15 -0.05 -0.67 -1.65
2 6 19 7 294 0.05 0.73 0.68 -0.69 -0.05 -0.67 -1.60
2 6 20 3 112 0.09 2.96 0.68 0.12 -0.05 0.43 -1.72
2 6 21 2 270 0.04 1.61 0.68 -0.71 -0.05 -0.67 -0.87
2 6 22 8 279 0.09 0.84 0.68 -0.36 -0.05 -0.67 -0.84
2 6 23 2 237 0.08 2.30 -0.49 0.15 -0.05 -0.67 -1.12
2 7 1 8 210 0.16 0.51 -1.18 0.46 1.37 0.43 1.10
2 7 2 16 231 0.12 1.01 -0.91 0.35 1.37 -0.67 0.40
2 7 3 12 211 0.1 1.39 -0.89 0.37 1.37 -0.67 0.29
2 7 4 8 209 0.06 0.35 -0.89 1.20 1.37 -0.67 0.35
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2 7 6 12 202 0.18 0.49 -1.02 0.55 1.37 0.43 0.36
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2 7 10 20 220 0.09 1.20 -0.91 0.45 1.37 -0.67 0.84
2 7 11 9 265 0.07 0.48 -1.27 0.45 1.37 -0.67 0.66
2 7 12 14 190 0.21 1.09 -0.89 0.28 1.37 0.43 0.63
2 7 13 9 222 0.09 0.82 -1.06 0.45 1.37 -0.67 -0.07
2 7 14 7 168 0.2 0.97 -1.00 0.03 1.37 0.43 -0.83
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2 7 17 7 262 0.04 0.22 -0.89 0.45 1.37 -0.67 0.04
2 7 18 7 285 0.07 0.28 -1.22 0.45 1.37 -0.67 1.12
2 7 19 6 179 0.12 1.31 -0.08 0.35 1.37 -0.67 -0.26
2 7 20 17 189 0.14 1.26 -0.08 0.46 1.37 -0.67 0.84
2 7 21 3 316 0.02 0.10 -1.27 0.45 1.37 -0.67 1.57
2 7 22 11 183 0.11 1.44 -0.08 0.34 1.37 -0.67 0.42
2 7 23 11 251 0.09 0.72 -1.24 0.29 1.37 -0.67 0.36
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2 8 2 2 282 0 -1.43 0.68 0.45 -0.76 0.43 -2.29
2 8 3 8 182 0.09 -0.47 2.30 0.49 -0.05 0.43 -0.85
2 8 4 10 236 0.19 -0.61 2.30 0.49 -0.05 -0.67 -1.01
2 8 5 13 87 0.15 -0.66 2.30 1.70 -0.05 -0.16 -2.28
2 8 6 4 117 0.13 -1.29 0.49 2.29 -0.76 -0.67 -2.17
2 8 7 2 64 0.02 -1.12 2.30 2.29 -0.76 -0.67 -2.37
2 8 8 3 104 0.06 -0.17 2.30 2.29 -0.05 -0.67 -0.98
2 8 9 4 170 0.08 -0.90 1.11 1.35 -0.05 0.43 -1.21
2 8 10 7 146 0.07 -0.91 2.30 0.47 -0.05 -0.67 -2.34
2 8 11 12 409 0.11 -0.78 2.30 -0.33 -0.76 -0.67 -1.03
2 8 12 7 241 0.1 -0.19 2.30 0.33 -0.05 -0.67 -0.89
2 8 13 4 258 0.09 0.04 2.30 -0.08 -0.05 -0.67 -1.04
2 8 14 9 330 0.09 -1.17 2.30 0.45 -0.76 -0.67 -2.26
2 8 15 6 212 0.13 -0.74 0.68 0.30 -0.05 0.43 -2.40
2 8 16 7 175 0.07 -0.53 2.30 0.49 -0.05 0.43 -1.23
2 8 17 4 109 0.04 -0.38 2.30 1.70 -0.05 0.43 -1.05
2 8 18 10 144 0.09 -0.89 0.68 1.70 -0.05 -0.67 -2.20
2 8 19 1 283 0 -1.59 0.68 -0.19 -0.05 0.43 -2.29
2 8 20 7 123 0.17 -0.94 2.30 0.46 -0.35 0.43 -2.27
2 8 21 3 329 0.06 -0.86 2.30 -0.29 -0.76 0.43 -0.98
2 8 22 21 277 0.11 -1.16 0.68 0.48 -0.05 -0.67 -2.25
2 8 23 0 NA NA NA NA NA NA NA NA
2 9 1 33 331 0.22 -0.57 -0.08 0.41 -0.05 -0.67 -0.71
2 9 2 10 368 0.19 -0.75 0.68 -0.60 -0.76 0.43 -0.42
2 9 3 13 423 0.19 -1.02 0.68 -0.69 -0.70 -0.67 -0.83
2 9 4 8 335 0.2 -0.08 0.68 -0.49 -0.23 -0.67 -0.86
2 9 5 15 377 0.21 -1.31 -0.08 0.30 -0.76 0.43 -0.98
2 9 6 8 224 0.04 -0.74 1.11 0.50 -0.05 0.43 -1.22
2 9 7 22 261 0.13 -0.22 0.68 0.29 -0.05 0.43 -0.60
2 9 8 14 381 0.28 -0.89 0.57 -0.09 -0.51 -0.67 -0.23
2 9 9 14 257 0.08 -0.64 0.68 0.49 -0.05 0.43 -0.76
2 9 10 13 354 0.27 -1.37 0.33 0.47 -0.27 -0.67 -0.93
2 9 11 9 248 0.16 -1.00 0.68 0.57 -0.05 0.43 -1.22
2 9 12 23 323 0.13 -0.51 0.68 0.02 -0.05 -0.67 -0.46
2 9 13 10 344 0.25 -1.39 0.68 -0.30 -0.48 0.43 -1.23
2 9 14 11 267 0.11 -1.03 0.68 0.41 -0.05 0.43 -0.85
2 9 15 20 422 0.17 -1.43 0.68 -0.23 -0.72 -0.67 -1.02
2 9 16 23 206 0.26 -0.52 0.70 1.25 -0.05 0.43 -0.79
2 9 17 8 304 0.13 -0.86 0.68 -0.12 -0.05 0.43 -0.39
2 9 18 8 240 0.05 -0.72 1.11 0.50 -0.05 0.43 -0.83
2 9 19 32 303 0.35 -0.82 0.68 0.33 0.04 -0.67 -1.07
2 9 20 13 317 0.16 -1.11 0.68 0.34 -0.76 0.43 -0.90
2 9 21 10 308 0.22 -0.75 -0.24 0.33 -0.05 0.43 -0.69
2 9 22 33 296 0.2 -0.39 0.68 0.55 -0.05 -0.67 -0.56
2 9 23 3 196 0.03 -0.83 -0.08 1.70 -0.05 -0.67 -1.98
2 10 1 4 269 0.07 0.51 0.68 -0.55 -0.05 0.43 -0.62
2 10 2 8 327 0.1 -0.43 0.68 -0.12 -0.05 0.43 1.46
2 10 3 23 318 0.15 0.23 -0.08 -0.23 -0.05 0.43 0.69
2 10 4 3 346 0.12 -0.04 0.68 -0.74 -0.76 0.43 -0.31
2 10 5 9 271 0.17 0.14 0.73 0.34 -0.05 0.43 1.57
2 10 6 10 273 0.12 0.92 0.68 -0.54 -0.05 0.43 0.74
2 10 7 9 299 0.11 0.93 -0.08 -0.36 -0.05 0.43 0.66
2 10 8 6 307 0.08 -0.01 0.68 -0.52 -0.05 0.43 0.38
2 10 9 8 289 0.15 0.67 -0.08 -0.27 -0.05 0.43 -0.15
2 10 10 13 325 0.26 0.16 -0.08 0.00 -0.16 0.43 1.23
2 10 11 13 315 0.18 -0.46 0.68 -0.14 -0.05 0.43 0.64
2 10 12 10 259 0.12 0.83 0.68 -0.14 -0.05 0.43 0.32
2 10 13 7 234 0.06 -0.20 0.68 1.23 -0.05 0.43 0.35
2 10 14 12 260 0.16 0.74 0.72 0.08 -0.05 0.43 0.89
2 10 15 8 249 0.12 0.66 0.68 0.48 -0.14 0.43 0.81
2 10 16 13 268 0.09 0.04 0.68 0.49 -0.05 0.43 0.60
2 10 17 11 256 0.11 0.31 0.68 0.24 -0.05 0.43 -0.06
2 10 18 6 342 0.17 0.82 0.30 -0.76 -0.76 0.43 0.64
2 10 19 8 302 0.15 0.22 0.68 -0.33 -0.05 0.43 1.20
2 10 20 12 370 0.24 -0.28 0.75 -0.51 -0.76 0.43 0.84
2 10 21 19 281 0.11 0.29 0.68 -0.07 -0.05 0.43 0.64
2 10 22 9 290 0.13 -0.34 0.68 0.15 -0.05 0.43 0.20
2 10 23 13 351 0.11 0.21 0.68 -0.70 -0.76 0.43 0.66
2 11 1 10 411 0.22 -1.09 -0.93 0.23 -0.12 -0.67 -0.72
2 11 2 14 502 0.14 -1.13 -0.97 -1.01 -0.86 -0.67 -0.09
2 11 3 7 481 0.2 -0.81 -0.08 -1.20 -1.11 -0.36 -0.96
2 11 4 19 516 0.16 -0.95 -0.95 -1.13 -0.91 -0.67 -1.64
2 11 5 14 482 0.12 -1.58 -0.89 -0.16 -0.76 -0.67 -0.68
2 11 6 24 498 0.16 -1.36 -0.94 -0.85 -0.77 -0.67 -0.46
2 11 7 17 527 0.25 -1.29 -1.25 -1.06 -0.84 -0.61 -1.65
2 11 8 23 484 0.15 -1.01 -0.92 -0.84 -0.79 -0.67 -0.72
2 11 9 18 509 0.1 -1.67 -0.91 -0.78 -0.76 -0.67 -0.75
2 11 10 10 435 0.15 -0.94 -0.08 -0.59 -0.69 -0.67 -0.89
2 11 11 19 491 0.25 -1.54 -0.93 -0.50 -0.74 -0.67 -0.11
2 11 12 12 452 0.25 -1.47 -0.76 -0.12 -0.79 0.43 -0.55
2 11 13 8 445 0.17 -1.22 -0.08 -0.43 -0.67 -0.67 -0.19
2 11 14 8 427 0.08 -0.60 -0.89 -0.75 -0.05 -0.67 -0.70
2 11 15 17 466 0.16 -0.52 -0.98 -0.84 -0.76 -0.67 -0.67
2 11 16 17 511 0.16 -1.09 -1.00 -1.06 -0.88 -0.67 -1.12
2 11 17 8 437 0.08 -0.69 -0.08 -0.75 -0.76 -0.67 -0.48
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2 11 19 10 525 0.18 -1.69 -1.19 -1.13 -0.90 -0.67 -0.23
2 11 20 30 473 0.29 -1.06 -0.87 -0.89 -0.79 0.43 -0.61
2 11 21 6 450 0.1 -1.62 -0.08 -0.06 -0.76 -0.67 -0.70
2 11 22 18 457 0.26 -1.05 -0.89 -0.10 -0.68 -0.67 -0.16
2 11 23 12 453 0.09 -1.43 -0.08 -0.29 -0.76 -0.67 -1.02
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2 12 2 13 394 0.18 -0.08 0.68 -0.81 -0.79 -0.67 -0.16
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2 12 5 15 442 0.11 -0.32 0.68 -1.19 -1.09 -0.67 0.31
2 12 6 12 458 0.26 -0.88 0.68 -0.73 -0.70 -0.67 1.24
2 12 7 17 405 0.25 -0.57 0.68 -0.86 -0.90 0.43 0.44
2 12 8 23 421 0.14 0.08 -0.08 -0.77 -0.76 -0.67 0.62
2 12 9 24 468 0.16 -0.66 -0.08 -1.13 -0.98 -0.67 0.37
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2 12 11 8 449 0.09 -0.21 0.84 -0.78 -0.76 -0.67 1.57
2 12 12 10 454 0.22 -0.75 0.72 -0.93 -0.94 0.43 1.25
2 12 13 19 428 0.17 -0.83 -0.08 -0.26 -0.68 -0.67 0.55
2 12 14 8 506 0.12 -0.51 1.05 -1.19 -1.11 -0.67 1.36
2 12 15 9 443 0.16 -0.77 -0.08 -0.08 -0.76 -0.67 1.22
2 12 16 21 441 0.19 -0.46 -0.08 -0.80 -0.62 -0.67 0.85
2 12 17 21 436 0.14 -0.85 0.68 -0.87 -0.76 -0.67 0.44
2 12 18 27 492 0.15 -0.84 -0.08 -1.17 -1.05 -0.67 0.90
2 12 19 9 495 0.15 -0.81 -0.08 -0.91 -0.88 -0.67 1.48
2 12 20 19 431 0.24 -0.99 0.68 -0.80 -0.78 -0.67 -0.34
2 12 21 16 387 0.2 -0.62 0.68 -0.11 -0.67 -0.67 0.49
2 12 22 10 477 0.19 -0.79 -0.08 -1.19 -1.11 0.43 0.74
2 12 23 27 415 0.13 -0.11 0.76 -0.76 -0.76 -0.67 1.01
2 13 1 10 138 0.21 0.94 0.68 0.67 1.37 0.43 -0.06
2 13 2 14 131 0.21 2.10 0.68 0.25 1.37 0.43 0.77
2 13 3 4 198 0.02 0.61 0.68 0.45 1.37 -0.67 0.69
2 13 4 2 47 0.19 1.96 1.49 2.29 1.37 0.43 1.36
2 13 5 8 143 0.08 1.28 1.11 0.45 1.37 -0.67 1.36
2 13 6 4 98 0.2 0.98 2.30 0.73 1.37 0.43 0.00
2 13 7 10 155 0.13 1.55 0.68 -0.28 1.37 0.43 0.33
2 13 8 24 151 0.12 1.45 0.68 0.22 1.37 0.43 0.72
2 13 9 5 164 0.11 1.30 -0.08 0.40 1.37 0.43 0.73
2 13 10 12 176 0.12 1.08 0.68 -0.15 1.37 0.43 0.72
2 13 11 10 94 0.13 1.39 2.30 0.44 1.37 0.43 1.23
2 13 12 12 154 0.12 1.75 0.68 0.36 1.37 -0.67 0.29
2 13 13 4 174 0.1 0.56 0.68 0.82 1.37 -0.67 0.19
2 13 14 18 153 0.27 0.89 0.75 0.30 1.37 0.43 1.33
2 13 15 10 172 0.22 0.58 0.77 0.34 1.37 0.43 0.47
2 13 16 8 177 0.08 0.88 0.68 0.45 1.37 -0.67 1.41
2 13 17 2 125 0.09 2.71 0.68 0.34 1.37 -0.67 0.67
2 13 18 8 95 0.07 1.62 2.30 0.45 1.37 -0.67 1.41
2 13 19 7 173 0.07 1.27 0.68 0.45 1.37 -0.67 0.74
2 13 20 7 128 0.1 1.78 0.68 0.38 1.37 0.43 -0.02
2 13 21 16 111 0.2 1.15 2.30 0.59 1.37 -0.67 0.18
2 13 22 10 152 0.11 1.78 0.68 0.46 1.37 -0.67 0.90
2 13 23 4 162 0.07 1.30 0.68 0.46 1.37 -0.67 0.05
2 14 1 46 475 0.08 -0.74 -0.89 -0.78 -0.76 -0.67 0.52
2 14 2 36 469 0.16 -0.24 -0.94 -0.83 -0.76 -0.67 0.94
2 14 3 19 439 0.18 -0.52 -0.95 -0.74 -0.05 -0.67 0.60
2 14 4 13 459 0.15 0.21 -1.04 -0.93 -0.76 -0.67 0.60
2 14 5 24 501 0.14 -0.59 -1.00 -1.16 -0.89 -0.67 0.87
2 14 6 44 480 0.13 -0.63 -0.93 -0.78 -0.76 -0.67 0.98
2 14 7 35 493 0.13 -0.68 -0.91 -1.17 -0.98 -0.67 0.42
2 14 8 19 451 0.1 -0.20 -0.95 -0.74 -0.76 -0.67 -0.06
2 14 9 19 483 0.14 -0.72 -1.03 -0.93 -0.85 -0.67 -0.07
2 14 10 34 488 0.13 -0.61 -1.27 -0.93 -0.76 -0.67 0.54
2 14 11 11 455 0.18 -0.85 -0.89 -0.17 -0.69 -0.67 0.45
2 14 12 37 462 0.1 -0.26 -0.91 -0.85 -0.76 -0.67 0.39
2 14 13 0 NA NA NA NA NA NA NA NA
2 14 14 0 NA NA NA NA NA NA NA NA
2 14 15 0 NA NA NA NA NA NA NA NA
2 14 16 0 NA NA NA NA NA NA NA NA
2 14 17 0 NA NA NA NA NA NA NA NA
2 14 18 0 NA NA NA NA NA NA NA NA
2 14 19 0 NA NA NA NA NA NA NA NA
2 14 20 0 NA NA NA NA NA NA NA NA
2 14 21 0 NA NA NA NA NA NA NA NA
2 14 22 0 NA NA NA NA NA NA NA NA
2 14 23 0 NA NA NA NA NA NA NA NA
2 15 1 13 341 0.17 0.19 -0.92 -0.52 -0.05 0.43 -0.93
2 15 2 16 361 0.16 0.24 0.68 -0.91 -0.45 -0.67 -1.62
2 15 3 5 456 0.11 -0.29 -0.89 -0.70 -0.76 -0.67 -1.12
2 15 4 26 461 0.2 -0.35 -0.08 -1.04 -0.81 -0.67 -1.59
2 15 5 16 444 0.3 -0.40 0.68 -1.15 -0.94 -0.61 -1.48
2 15 6 15 486 0.08 -0.50 -0.89 -0.92 -0.76 -0.67 -1.66
2 15 7 5 203 0.2 0.69 -1.12 -0.43 1.37 -0.23 -1.58
2 15 8 9 426 0.33 -0.40 -0.75 -0.77 -0.68 0.43 -1.23
2 15 9 14 369 0.19 0.28 -0.92 -0.38 -0.05 -0.67 -0.90
2 15 10 11 464 0.07 -0.06 -0.89 -0.82 -0.76 -0.67 -1.66
2 15 11 5 460 0.11 0.54 -0.89 -0.88 -0.76 -0.67 -1.70
2 15 12 7 410 0.17 0.51 -0.08 -0.92 -0.76 -0.67 -1.57
2 15 13 17 352 0.15 0.35 -0.89 -0.72 -0.13 0.43 -1.66
2 15 14 11 420 0.08 0.02 -1.03 -0.67 -0.05 -0.67 -1.65
2 15 15 13 396 0.13 0.43 -0.89 -0.71 -0.05 -0.67 -1.63
2 15 16 4 379 0.23 0.75 -0.69 -0.65 -0.76 0.15 -0.67
2 15 17 7 322 0.21 0.47 0.24 -0.72 -0.66 0.43 -1.52
2 15 18 25 343 0.22 0.36 -0.08 -0.63 -0.05 -0.67 -1.53
2 15 19 15 496 0.09 -0.38 -1.04 -1.11 -0.76 -0.67 -1.63
2 15 20 14 380 0.3 0.03 -0.08 -0.55 -0.46 -0.67 -0.82
2 15 21 14 293 0.21 0.50 -0.08 -0.67 -0.05 0.43 -1.14
2 15 22 10 353 0.17 0.20 0.68 -0.81 -0.48 -0.67 -0.92
2 15 23 11 432 0.09 -0.35 -0.89 -0.80 -0.05 -0.67 -1.61
2 16 1 8 58 0.25 0.04 2.30 1.70 -0.05 2.64 -1.20
2 16 2 1 33 0 1.25 2.30 1.20 1.37 2.64 -0.94
2 16 3 13 90 0.39 0.07 2.30 0.48 -0.05 2.64 -0.99
2 16 4 5 230 0.11 0.87 -0.89 -0.67 -0.33 2.64 -1.64
2 16 5 11 163 0.15 -0.13 0.68 0.38 -0.05 2.64 -0.78
2 16 6 4 75 0.04 0.84 -0.89 1.20 1.37 2.64 -0.83
2 16 7 17 207 0.2 -0.62 0.68 0.36 -0.76 2.64 -1.01
2 16 8 20 135 0.14 -0.24 0.98 0.51 -0.05 2.64 -1.10
2 16 9 9 132 0.15 1.33 -0.08 -0.65 -0.29 2.64 -1.52
2 16 10 8 74 0.2 0.95 0.39 1.10 1.37 2.64 -0.65
2 16 11 8 284 0.05 -0.76 -0.08 0.51 -0.76 2.64 -1.16
2 16 12 5 83 0.04 0.78 -0.89 1.20 1.37 2.64 -0.43
2 16 13 5 105 0.07 -0.44 0.68 0.49 -0.05 2.64 -2.23
2 16 14 4 136 0.09 1.24 0.68 -0.64 -0.05 2.64 -0.87
2 16 15 2 106 0.03 1.56 0.68 -0.71 -0.76 2.64 -1.71
2 16 16 9 120 0.06 -0.01 0.68 1.23 -0.05 2.64 -0.72
2 16 17 7 67 0.21 0.47 0.41 1.49 1.37 2.64 -0.97
2 16 18 7 213 0.33 0.07 0.57 -0.21 -0.05 2.64 -0.36
2 16 19 5 160 0.15 0.43 0.68 0.62 -0.05 2.64 0.17
2 16 20 6 320 0.13 -0.58 0.68 -0.26 -0.76 2.64 -0.44
2 16 21 3 85 0.06 0.38 0.68 2.29 -0.05 2.64 -0.61
2 16 22 6 345 0.1 -0.82 -0.08 0.21 -0.76 2.64 -0.79
2 16 23 5 110 0.06 1.39 0.68 -0.67 -0.05 2.64 -1.61
2 17 1 6 440 0.09 -0.58 -1.27 0.12 -0.05 -0.67 1.29
2 17 2 10 416 0.13 -0.58 -0.93 -0.18 -0.05 -0.67 0.64
2 17 3 12 417 0.09 -0.24 -0.92 0.15 -0.05 -0.67 1.57
2 17 4 9 349 0.12 0.29 -0.93 0.23 -0.05 -0.67 -0.11
2 17 5 5 309 0.28 0.52 -0.89 0.35 -0.05 0.43 0.52
2 17 6 16 385 0.13 0.31 -0.89 -0.45 -0.05 -0.67 0.41
2 17 7 23 372 0.09 0.31 -0.91 0.03 -0.05 -0.67 0.86
2 17 8 21 403 0.09 -0.15 -1.27 0.09 -0.05 -0.67 0.85
2 17 9 14 366 0.13 -0.37 -0.92 0.45 -0.05 -0.67 -0.31
2 17 10 11 419 0.13 -0.26 -0.89 -0.36 -0.05 -0.67 1.02
2 17 11 13 389 0.11 -0.35 -0.95 0.02 -0.05 -0.67 0.00
2 17 12 17 392 0.06 -0.26 -0.89 0.05 -0.05 -0.67 0.62
2 17 13 4 362 0.17 0.15 -1.08 0.01 -0.05 -0.67 -0.67
2 17 14 9 424 0.17 0.37 -0.89 -0.29 -0.76 -0.67 0.90
2 17 15 12 395 0.09 -0.06 -1.27 -0.09 -0.05 -0.67 0.37
2 17 16 18 399 0.11 -0.17 -0.89 0.09 -0.05 -0.67 1.05
2 17 17 15 373 0.09 -0.59 -0.89 0.45 -0.05 -0.67 -0.58
2 17 18 11 357 0.09 0.20 -0.89 0.47 -0.05 -0.67 0.68
2 17 19 15 398 0.26 0.98 -0.89 -0.41 -0.57 -0.67 0.49
2 17 20 20 367 0.08 0.25 -0.91 0.05 -0.05 -0.67 0.46
2 17 21 3 438 0.08 -0.60 -0.89 0.04 -0.76 -0.67 0.48
2 17 22 6 429 0.08 -0.22 -0.89 -0.07 -0.76 -0.67 0.68
2 17 23 15 382 0.08 -0.35 -0.92 0.45 -0.05 -0.67 0.47
2 18 1 4 430 0.06 -0.12 2.30 -0.78 -0.76 -0.67 0.52
2 18 2 5 199 0.07 0.53 2.30 0.46 -0.05 0.43 0.62
2 18 3 4 276 0.04 -0.21 2.30 0.03 -0.05 -0.67 -0.48
2 18 4 1 225 0 0.74 2.30 -0.19 -0.05 -0.67 -0.79
2 18 5 4 340 0.08 0.42 2.30 -0.56 -0.76 -0.67 -0.64
2 18 6 16 286 0.15 0.56 2.30 0.00 -0.05 -0.67 1.49
2 18 7 5 242 0.04 0.41 2.30 0.45 -0.05 -0.67 0.35
2 18 8 7 187 0.09 0.61 2.30 0.31 -0.05 0.43 1.57
2 18 9 8 200 0.15 0.62 2.30 0.08 -0.05 0.43 0.04
2 18 10 5 226 0.04 0.41 2.30 0.05 -0.05 0.43 0.69
2 18 11 8 463 0.07 0.24 2.30 -0.78 -0.76 -0.67 1.30
2 18 12 9 522 0.08 -0.14 2.30 -1.19 -1.11 -0.67 1.33
2 18 13 2 115 0.08 1.74 2.30 1.23 -0.05 -0.67 -0.03
2 18 14 6 274 0.18 0.31 2.30 -0.06 -0.05 -0.67 0.15
2 18 15 3 229 0.06 0.55 2.30 -0.06 -0.05 0.43 1.14
2 18 16 10 254 0.13 0.67 2.30 0.22 -0.05 -0.67 0.87
2 18 17 4 407 0.13 0.00 2.30 -0.78 -0.76 0.43 0.94
2 18 18 7 246 0.08 0.16 2.30 -0.04 -0.05 0.43 1.57
2 18 19 4 339 0.04 -0.04 2.30 0.05 -0.76 -0.67 0.04
2 18 20 3 217 0.01 0.65 2.30 0.03 -0.05 0.43 0.69
2 18 21 3 476 0.14 -0.52 2.30 -0.45 -0.76 -0.67 1.29
2 18 22 0 NA NA NA NA NA NA NA NA
2 18 23 0 NA NA NA NA NA NA NA NA
2 19 1 6 391 0.18 0.52 -0.89 -0.34 -0.64 0.43 0.98
2 19 2 6 383 0.07 0.68 -0.89 -0.56 -0.76 0.43 0.10
2 19 3 21 470 0.12 -1.02 -0.91 -0.76 -0.76 0.43 0.53
2 19 4 11 348 0.2 0.08 -1.03 -0.10 -0.05 0.43 0.14
2 19 5 10 446 0.15 -1.12 -0.81 -0.30 -0.76 0.43 0.01
2 19 6 14 350 0.08 0.07 -0.89 0.05 -0.05 0.43 0.74
2 19 7 8 388 0.27 -0.53 -0.38 -0.56 -0.40 0.43 -0.28
2 19 8 7 467 0.11 -1.18 -0.89 -0.29 -0.76 0.43 0.88
2 19 9 6 425 0.09 -0.49 -0.89 -0.23 -0.76 0.43 0.58
2 19 10 9 406 0.1 -0.18 -0.08 -0.75 -0.76 0.43 0.63
2 19 11 10 400 0.16 -0.36 -0.89 -0.71 -0.05 0.43 0.33
2 19 12 9 347 0.21 -0.22 -0.89 0.49 -0.13 0.43 0.63
2 19 13 8 497 0.1 -0.83 -0.89 -1.19 -1.11 0.43 0.44
2 19 14 16 378 0.12 -0.25 -1.03 -0.12 -0.05 0.43 0.74
2 19 15 16 418 0.15 -0.73 -0.08 -0.41 -0.76 0.43 0.82
2 19 16 22 448 0.15 -0.21 -0.94 -0.82 -0.76 0.43 0.77
2 19 17 13 485 0.15 -0.74 -0.95 -0.83 -0.76 0.43 1.20
2 19 18 13 447 0.14 -0.52 -0.98 -0.75 -0.76 0.43 0.17
2 19 19 10 412 0.21 -0.48 -0.81 -0.43 -0.05 0.43 1.25
2 19 20 11 355 0.16 -0.61 -0.89 0.20 -0.05 0.43 -0.14
2 19 21 9 390 0.1 -0.66 -0.89 -0.14 -0.05 0.43 0.65
2 19 22 10 401 0.15 -0.52 -0.93 0.13 -0.05 0.43 1.40
2 19 23 11 487 0.12 -0.83 -1.27 -0.86 -0.76 0.43 0.75
2 20 1 5 73 0.02 -0.13 0.68 1.70 1.37 0.43 -2.46
2 20 2 3 69 0.02 -0.02 0.68 2.29 1.37 0.43 -1.98
2 20 3 3 65 0.19 -0.83 1.22 3.07 -0.05 -0.67 -2.46
2 20 4 7 79 0.03 0.82 2.30 1.21 1.37 0.43 -0.92
2 20 5 5 62 0.03 0.13 0.68 2.29 1.37 0.43 -2.32
2 20 6 3 80 0.04 -0.03 0.68 1.70 1.37 0.43 -2.19
2 20 7 3 71 0.03 0.47 2.30 1.70 1.37 0.43 -0.79
2 20 8 2 72 0 0.22 0.68 1.70 1.37 0.43 -2.46
2 20 9 12 35 0.14 0.15 2.30 2.29 1.37 0.43 -2.29
2 20 10 4 44 0.06 -0.03 0.68 3.07 1.37 0.43 -2.25
2 20 11 11 42 0.07 0.15 2.30 1.70 1.37 0.43 -2.39
2 20 12 1 31 0 0.57 2.30 3.30 1.37 0.43 -1.25
2 20 13 5 63 0.08 -0.29 0.68 2.29 1.37 0.43 -2.35
2 20 14 6 66 0.05 0.39 2.30 1.70 1.37 0.43 -1.23
2 20 15 1 68 0 0.31 0.68 2.29 1.37 0.43 -1.98
2 20 16 3 78 0.05 1.06 2.30 1.23 1.37 0.43 -0.62
2 20 17 4 27 0.05 0.19 2.30 3.07 1.37 0.43 -2.25
2 20 18 0 NA NA NA NA NA NA NA NA
2 20 19 0 NA NA NA NA NA NA NA NA
2 20 20 0 NA NA NA NA NA NA NA NA
2 20 21 0 NA NA NA NA NA NA NA NA
2 20 22 0 NA NA NA NA NA NA NA NA
2 20 23 0 NA NA NA NA NA NA NA NA
2 21 1 4 504 0.17 0.72 -0.89 -0.66 -0.76 2.64 1.36
2 21 2 5 245 0.17 0.97 0.68 -0.65 -0.62 2.64 1.03
2 21 3 6 313 0.16 -0.20 0.68 -0.40 -0.52 2.64 0.41
2 21 4 10 338 0.18 0.27 -0.89 -0.34 -0.05 2.64 0.32
2 21 5 4 375 0.13 0.16 -0.49 0.04 -0.76 2.64 0.66
2 21 6 6 298 0.15 1.35 -0.08 -0.45 -0.76 2.64 0.94
2 21 7 10 526 0.21 -0.31 -1.08 -0.88 -0.76 2.64 1.28
2 21 8 11 490 0.15 -0.83 -0.89 -0.25 -0.76 2.64 -0.33
2 21 9 6 393 0.19 1.39 -0.89 -0.56 -0.76 2.64 -0.04
2 21 10 8 359 0.11 0.49 -0.89 -0.39 -0.05 2.64 0.95
2 21 11 6 295 0.1 0.19 -0.08 -0.29 -0.05 2.64 0.75
2 21 12 7 376 0.09 -0.05 -0.89 -0.19 -0.05 2.64 0.77
2 21 13 18 205 0.2 0.58 0.68 -0.13 -0.13 2.64 0.65
2 21 14 6 186 0.18 0.55 0.55 0.01 -0.05 2.64 1.50
2 21 15 6 328 0.14 0.27 -0.89 -0.73 -0.05 2.64 -0.76
2 21 16 2 215 0.1 0.75 -0.08 -0.46 -0.05 2.64 -0.35
2 21 17 5 408 0.12 -0.53 -0.08 -0.29 -0.76 2.64 0.19

Now let’s check the compression summary for HVT (map C). The table below shows no of cells, no of cells having quantization error below threshold and percentage of cells having quantization error below threshold for each level.

mapC_compression_summary <- map_C[[3]]$compression_summary %>%  dplyr::mutate_if(is.numeric, funs(round(.,4)))
compressionSummaryTable(mapC_compression_summary)
segmentLevel noOfCells noOfCellsBelowQuantizationError percentOfCellsBelowQuantizationErrorThreshold parameters
1 23 0 0 n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans
2 508 434 0.85 n_cells: 23 quant.err: 0.2 distance_metric: L1_Norm error_metric: max quant_method: kmeans

As it can be seen from the table above, 0% of the cells have hit the quantization threshold error in level 1 and 85% of the cells have hit the quantization threshold error in level 2

Let’s plot the Voronoi tessellation for layer 2 (map C)

HVT::plotHVT(map_C,
        line.width = c(0.4,0.2), 
        color.vec = c("#141B41","#0582CA"),
        centroid.size = 0.1,
        maxDepth = 2) 
Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Figure 14: The Voronoi Tessellation for layer 2 (map C) shown for the 100 cells in the dataset ’computers’ at level 2

Heat Maps

Now let’s plot all the features for each cell at level two as a heatmap for better visualization.

The heatmaps displayed below provides a visual representation of the spatial characteristics of the computers data, allowing us to observe patterns and trends in the distribution of each of the features (n,price,speed,hd,ram,screen,ads). The sheer green shades highlight regions with higher values in each of the heatmaps, while the indigo shades indicate areas with the lowest values in each of the heatmaps. By analyzing these heatmaps, we can gain insights into the variations and relationships between each of these features within the computers data.


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "n",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 15: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell

Figure 15: The Voronoi Tessellation with the heat map overlaid for features No. of entities in each cell


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "price",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 16: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset

Figure 16: The Voronoi Tessellation with the heat map overlaid for features price in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "speed",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 17: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset

Figure 17: The Voronoi Tessellation with the heat map overlaid for features speed in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "hd",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 18: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset

Figure 18: The Voronoi Tessellation with the heat map overlaid for features hd in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ram",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 19: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset

Figure 19: The Voronoi Tessellation with the heat map overlaid for features ram in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "screen",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 20: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset

Figure 20: The Voronoi Tessellation with the heat map overlaid for features screen in the ’computers’ dataset


  hvtHmap(
  map_C,
  trainComputers,
  child.level = 2,
  hmap.cols = "ads",
  line.width = c(0.6,0.4),
  color.vec = c("#141B41","#0582CA"),
  palette.color = 6,
  centroid.size = 0.1,
  show.points = T,
  quant.error.hmap = 0.2,
  n_cells.hmap = 100,
) 
Figure 21: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

Figure 21: The Voronoi Tessellation with the heat map overlaid for features ads in the ’computers’ dataset

We now have the set of maps (map A, map B & map C) which will be used to predict which map and cell each test record is assigned to, but before that lets view our test dataset

6 Prediction on Test Data

Now once we have built the model, let us try to predict using our test dataset (containing 1252 data points) which cell and which layer each point belongs to.

Raw Testing Dataset

The testing dataset includes the following columns:

Let’s have a look at our randomly selected test dataset containing 1252 datapoints.

Table(head(testComputers_data))
Row.No price speed hd ram screen ads
3 1595 25 170 4 15 94
4 1849 25 170 8 14 94
7 1720 25 170 4 14 94
10 2575 50 210 4 15 94
11 2195 33 170 8 15 94
14 2295 25 245 8 14 94

The predictLayerHVT function is used to score the test data using the predictive set of maps. This function takes an input - a test data and a set of maps (map A, map B, map C).

Now, Let us understand the predictLayerHVT function.

predictLayerHVT(data,
                map_A,
                map_B,
                map_C,
                mad.threshold = 0.2,
                normalize = T, 
                distance_metric="L1_Norm",
                error_metric="max",
                child.level = 1, 
                line.width = c(0.6, 0.4, 0.2),
                color.vec = c("#141B41", "#6369D1", "#D8D2E1"),
                yVar= NULL,
                ...)

Each of the parameters of predictLayerHVT function has been explained below:

The function predicts based on the HVT maps - map A, map B and map C, constructed using HVT function. For each test record, the function will assign that record to Layer1 or Layer2. Layer1 contains the cell ids from map A and Layer 2 contains cell ids from map B (novelty map) and map C (map without novelty).

Prediction Algorithm

The prediction algorithm recursively calculates the distance between each point in the test dataset and the cell centroids for each level. The following steps explain the prediction method for a single point in the test dataset:

  1. Calculate the distance between the point and the centroid of all the cells in the first level.
  2. Find the cell whose centroid has minimum distance to the point.
  3. Check if the cell drills down further to form more cells.
  4. If it doesn’t, return the path. Or else repeat steps 1 to 4 till we reach a level at which the cell doesn’t drill down further.

Note : The prediction algorithm will not work if some of the variables used to perform quantization are missing. In the test dataset, we should not remove any features


validation_data <- testComputers
new_predict <- predictLayerHVT(
    data=validation_data,
    map_A,
    map_B,
    map_C,
    normalize = T
  )

Let’s see which cell and layer each point belongs to and check the Mean Absolute Difference for each of the 1252 records.


act_pred <- new_predict[["actual_predictedTable"]]
rownames(act_pred) <- NULL
act_pred %>% head(1000) %>%as.data.frame() %>%Table(scroll = T)
Row.Number act_price act_speed act_hd act_ram act_screen act_ads Layer1.Cell.ID Layer2.Cell.ID pred_price pred_speed pred_hd pred_ram pred_screen pred_ads diff
1 -1.0786 -1.2710 -0.9473 -0.7590 0.4307 -1.7213 A24 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4690761
2 -0.6387 -1.2710 -0.9473 -0.0501 -0.6741 -1.7213 A142 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4423203
3 -0.8621 -1.2710 -0.9473 -0.7590 -0.6741 -1.7213 A29 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3712412
4 0.6187 -0.0817 -0.7914 -0.7590 0.4307 -1.7213 A237 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3987216
5 -0.0394 -0.8905 -0.9473 -0.0501 0.4307 -1.7213 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3772698
6 0.1338 -1.2710 -0.6551 -0.0501 -0.6741 -1.7213 A162 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3302619
7 0.8334 -0.0817 -0.7836 -0.0501 -0.6741 -1.7213 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3328709
8 -0.2126 -0.8905 -0.6356 -0.7590 0.4307 -1.7213 A165 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4049180
9 0.9997 0.6795 -1.1031 -0.7590 -0.6741 -1.7213 A214 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.5427887
10 1.1382 -0.0817 -0.7914 -0.7590 2.6404 -1.7213 A379 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.7270886
11 3.0780 -0.8905 0.1513 -0.0501 -0.6741 -1.7213 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.7243132
12 1.5193 -0.8905 -0.2850 1.3676 -0.6741 -1.7213 A316 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5546289
13 0.6533 -0.8905 -0.7914 -0.0501 2.6404 -1.7213 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.6569314
14 0.3243 -0.0817 -0.7914 -0.0501 -0.6741 -1.7213 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2493209
15 0.3589 -0.0817 -0.7914 -0.0501 -0.6741 -1.7213 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2550875
16 0.4871 -0.8905 -0.7836 -0.0501 -0.6741 -1.7213 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3049983
17 0.8265 -0.8905 -0.6551 -0.0501 -0.6741 -1.7213 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3822952
18 -0.8119 -1.2710 -1.1420 -0.7590 -0.6741 -1.7213 A29 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4120579
19 1.3461 0.6795 -0.2850 -0.0501 0.4307 -1.7213 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3298423
20 -0.7322 -0.8905 -0.9473 -0.7590 0.4307 -1.7213 A24 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4633928
21 -0.9054 -0.8905 -0.9473 -0.7590 -0.6741 -1.7213 A42 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3006079
22 1.4309 -0.0817 -0.6551 -0.0501 -0.6741 -1.7213 A215 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3679176
23 -1.0197 -1.2710 -1.2979 -0.0501 -0.6741 -1.7213 A10 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.5204834
24 0.6533 -1.2710 -0.6551 -0.0501 -0.6741 -1.7213 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4168452
25 -0.3789 -0.8905 -1.1420 -0.0501 -0.6741 -1.7213 A142 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3680537
26 0.3502 -0.8905 -0.9473 -0.0501 0.4307 -1.7213 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4077145
27 0.6533 -1.2710 -0.2850 1.3676 -0.6741 -1.7079 A316 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.6314504
28 -0.9054 -0.8905 -0.9473 -0.7590 -0.6741 -1.7079 A42 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2983746
29 1.3530 0.6795 -0.3240 -0.7590 0.4307 -1.7079 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4511090
30 1.3461 0.6795 -0.6356 -0.0501 2.6404 -1.7079 A386 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.5181657
31 -0.1260 0.6795 -0.9473 -0.7590 -0.6741 -1.7079 A166 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3903275
32 1.3461 0.6795 -0.2850 -0.0501 0.4307 -1.7079 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3276090
33 -1.2448 -1.2710 -0.9473 -0.7590 -0.6741 -1.7079 A10 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3258541
34 0.1857 0.6795 -0.6356 -0.0501 -0.6741 -1.7079 A250 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3735345
35 0.0039 -0.8905 -0.6356 -0.7590 -0.6741 -1.7079 A121 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2683518
36 1.3530 -0.0817 -0.9473 -0.7590 -0.6741 -1.7079 A214 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4466054
37 -0.0394 -0.8905 -0.9473 -0.0501 0.4307 -1.7079 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3750365
38 2.0458 0.6795 -0.7135 -0.7590 0.4307 -1.7079 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5576488
39 0.6602 -0.0817 -0.7914 -0.7590 -0.6741 -1.7079 A214 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3051554
40 0.9114 0.6795 -0.6551 -0.0501 -0.6741 -1.7079 A250 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4014226
41 2.7316 -0.8905 0.1513 -0.0501 -0.6741 -1.7079 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.6643465
42 0.9131 0.6795 -0.6356 -0.7590 0.4307 -1.7079 A292 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5763590
43 0.3139 -0.0817 -0.7836 -0.7590 -0.6741 -1.7079 A126 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2461387
44 0.7382 -0.0817 -0.6551 -0.0501 -0.6741 -1.7079 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3355012
45 0.6187 0.6795 -0.6356 -0.0501 -0.6741 -1.7079 A250 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4457012
46 0.8265 -0.8905 -0.2850 1.3676 -0.6741 -1.7079 A316 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.5391671
47 1.3530 0.6795 -0.6551 1.3676 0.4307 -1.6406 A370 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5052535
48 0.9824 -0.8905 -0.6356 -0.0501 2.6404 -1.6406 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.6723648
49 -1.0786 -0.8905 -1.2784 -1.1134 -0.6741 -1.6406 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3725412
50 2.7316 -0.8905 0.1513 -0.0501 -0.6741 -1.6406 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.6531299
51 2.9048 0.6795 0.3383 -0.0501 0.4307 -1.6406 A382 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5878379
52 0.7226 -0.8905 -0.6356 -0.0501 2.6404 -1.6406 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.6290648
53 0.6602 -0.0817 -0.7836 -0.0501 -0.6741 -1.6406 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2905542
54 0.9131 0.6795 -0.9473 -0.0501 -0.6741 -1.6406 A250 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4386226
55 1.1729 -0.0817 -0.2850 1.3676 -0.6741 -1.6406 A366 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4641122
56 -0.3789 -0.8905 -0.9473 -0.7590 -0.6741 -1.6406 A85 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3242382
57 -0.0394 -1.2710 -0.6551 -0.0501 -0.6741 -1.6406 A162 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3224339
58 -1.0786 -1.2710 -0.9473 -0.7590 0.4307 -1.6406 A24 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4556261
59 0.6187 0.6795 -0.6356 -0.0501 -0.6741 -1.6406 A250 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4344845
60 1.8726 0.6795 -0.6551 1.3676 0.4307 -1.6406 A370 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5180100
61 0.8334 0.6795 -0.7798 -0.0501 -0.6741 -1.6406 A250 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4239892
62 -0.8621 -1.2710 -0.9473 -0.7590 -0.6741 -1.6406 A29 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3577912
63 -0.3858 -0.8905 -0.6356 -0.7590 -0.6741 -1.6406 A85 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3220851
64 0.1407 -0.0817 -0.7836 -0.7590 -0.6741 -1.6406 A126 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2060554
65 0.5667 0.6795 -0.6356 -0.0501 0.4307 -1.6406 A292 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5047256
66 0.1164 -0.0817 -0.6356 -0.0501 -0.6741 -1.5331 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2059845
67 2.7316 -0.8905 0.1513 -0.0501 -0.6741 -1.5331 A367 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.6352132
68 0.6602 -0.0817 -0.7836 -0.0501 -0.6741 -1.5331 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2726375
69 1.1729 -0.0817 -0.2850 1.3676 -0.6741 -1.5331 A366 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4461955
70 -0.0394 0.6795 -0.9473 -0.7590 -0.6741 -1.5331 A166 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3467608
71 -0.9140 -0.8905 -1.2784 -1.1134 -0.6741 -1.5331 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3820579
72 1.5106 0.6795 -0.2850 1.3676 -0.6741 -1.5331 A366 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3390671
73 0.6602 -0.8905 -0.6746 -0.7590 0.4307 -1.5331 A165 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4202959
74 -0.0394 -0.0817 -1.1031 -0.7590 -0.6741 -1.5331 A126 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2458608
75 0.6533 -0.0817 -0.6356 -0.0501 0.4307 -1.5331 A261 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3937173
76 0.5148 -0.8905 -0.6356 -0.0501 0.4307 -1.5331 A223 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4004781
77 -0.2473 0.6795 -0.9473 -0.7590 -0.6741 -1.5331 A166 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3814108
78 1.0066 -0.0817 -0.6746 -0.7590 0.4307 -1.5331 A237 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4481852
79 0.1338 -0.8905 -0.6551 -0.0501 -0.6741 -1.5331 A162 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2354786
80 -1.2604 -1.2710 -1.2784 -1.1134 -0.6741 -1.5331 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4135708
81 -1.4249 -1.2710 -1.2784 -1.1134 -0.6741 -1.5331 A6 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4409875
82 0.4005 -0.8905 -0.7135 -0.7590 -0.6741 -1.5331 A121 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2722798
83 -0.5521 -0.8905 -0.7836 -0.7590 -0.6741 -1.1163 A104 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2313746
84 1.1642 -0.8905 0.1513 1.3676 -0.6741 -1.1163 A301 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.4148116
85 -0.0481 -0.8905 -0.7759 -0.7590 0.4307 -1.1163 A165 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3532458
86 -0.3858 -0.8905 -0.6356 -0.0501 -0.6741 -1.1163 A141 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3325784
87 0.4801 -0.0817 -0.6356 -0.0501 0.4307 -1.1163 A261 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3953451
88 0.3243 -0.0817 -0.6356 -0.0501 -0.6741 -1.1163 A215 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2711290
89 0.1338 -1.2710 -0.2850 -0.0501 -0.6741 -1.1163 A201 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3910731
90 1.0170 -0.8905 -0.6356 -0.0501 2.6404 -1.1163 A343 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.5907481
91 0.6602 -0.8905 -0.6551 1.3676 0.4307 -1.1163 A345 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.6451507
92 0.3069 -0.8905 -0.6356 -0.0501 -0.6741 -1.1163 A187 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2980731
93 0.3589 0.6795 -0.6356 -0.0501 0.4307 -1.1163 A292 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.4502633
94 0.5148 -0.8905 -0.6356 -0.0501 0.4307 -1.1163 A226 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4309725
95 2.2120 0.6795 0.3383 -0.0501 -0.6741 -1.1163 A337 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3873014
96 -0.3165 -0.8905 -0.6356 -0.0501 -0.6741 -1.1163 A141 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.3210284
97 0.6602 -0.0817 -0.0318 -0.7590 0.4307 -1.1163 A237 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.5280797
98 -0.0394 -0.0817 -0.7759 -0.7590 -0.6741 -1.1163 A152 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2237022
99 0.8178 -0.0817 -0.2850 -0.0501 0.4307 -1.1163 A261 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.4305373
100 1.3374 -0.0817 0.1513 1.3676 0.4307 -1.1163 A370 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.5289043
101 -0.0481 -0.8905 -0.7759 -0.7590 -0.6741 -1.1163 A104 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2549963
102 -0.1260 -0.0817 -0.9473 -0.7590 -0.6741 -1.1163 A152 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.2648219
103 0.8178 -0.0817 -0.2850 -0.0501 -0.6741 -1.1163 A228 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3702176
104 1.6924 0.6795 0.4941 -0.0501 0.4307 -1.1163 A330 C201 1.5745305 0.4514353 -0.2367672 0.2737663 -0.3026593 -1.3041998 0.3869878
105 0.7399 -0.0817 -0.6356 -0.0501 0.4307 -1.1163 A261 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4386451
106 -0.3789 -0.8905 -0.9473 -0.7590 -0.6741 -1.1163 A104 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2875246
107 1.2422 -0.0817 -0.6356 -0.0501 2.6404 -1.1163 A379 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.5235326
108 0.6187 -0.0817 -0.6356 -0.0501 0.4307 -1.1163 A261 C404 0.0640661 -0.3965677 -0.7815408 -0.3982964 -0.4164485 -1.4161834 0.4184451
109 -0.5521 -0.8905 -0.9473 -0.7590 -0.6741 -0.6189 A107 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2009794
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899 1.1642 0.6795 1.1953 1.3676 0.4307 -0.2290 A378 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.4439916
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903 -0.0325 0.6795 0.4473 -0.0501 0.4307 -0.2290 A287 C297 0.2458726 0.5046167 -0.1059150 -0.1622064 0.4307274 0.6555503 0.3338592
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906 1.9592 2.2969 1.2264 -0.0501 -0.6741 -0.2290 A393 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.7509662
907 1.5348 -0.8905 2.2860 2.7854 0.4307 -0.2290 A416 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.2451414
908 2.9619 2.2969 1.2264 -0.0501 0.4307 -0.2290 A393 C219 1.9755430 0.5864499 0.2031375 -0.2014525 0.1203785 0.5809108 0.8319424
909 1.0776 -0.8905 1.1953 1.3676 2.6404 -0.2290 A401 C228 0.8544662 -0.8230415 0.4195786 1.3676416 -0.4672508 0.5202082 0.8205357
910 -0.6543 -0.0817 -0.7759 -0.7590 0.4307 -0.2290 A164 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.3980366
911 -1.9376 -0.8905 -0.2850 -0.7590 -0.6741 -0.2290 A81 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3145551
912 -1.2448 -0.8905 -0.7914 -0.7590 -0.6741 -0.2290 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1451590
913 -0.8829 0.6795 0.4941 -0.7590 -0.6741 -0.2290 A182 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3434682
914 1.0863 -0.8905 2.2860 2.7854 0.4307 -0.2290 A416 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3019804
915 -0.2421 -0.8905 -0.2850 -0.7590 2.6404 -0.2290 A13 C360 0.2356188 -0.4603485 -0.3744382 -0.4107768 2.6404120 0.5948051 0.3615581
916 0.5477 0.6795 1.2342 -0.0501 2.6404 -0.2290 A387 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.3253056
917 -1.4249 -1.2710 -0.2850 -0.0501 -0.6741 -0.2290 A124 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.4095972
918 -0.2057 -0.8905 -0.7836 -0.0501 -0.6741 -0.2290 A133 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3213452
919 3.6893 2.2969 1.2264 -0.0501 2.6404 -0.2290 A434 C96 2.0782294 0.7478725 0.1122044 0.7127714 2.6404120 0.5611990 0.9712293
920 -1.1496 -0.8905 -0.7759 -0.7590 -0.6741 -0.2290 A93 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1378128
921 -0.4724 0.6795 -0.2850 -0.7590 -0.6741 -0.2290 A174 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3259140
922 -0.5677 -0.8905 0.4473 -0.0501 -0.6741 -0.8071 A186 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.4051200
923 -0.6906 -0.8905 -0.5577 -0.7590 -0.6741 -0.8071 A74 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1665373
924 -0.3079 -0.0817 0.4473 -0.0501 0.4307 -0.8071 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3429258
925 0.0385 -0.8905 2.2860 -0.0501 -0.6741 -0.8071 A258 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.6353449
926 0.0385 -0.0817 2.2860 -0.0501 -0.6741 -0.8071 A319 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.5137818
927 -0.9054 0.6795 -0.7759 -0.7590 -0.6741 -0.8071 A150 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3486013
928 0.6499 -0.0817 1.1953 1.3676 0.4307 -0.8071 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2784382
929 -0.3702 -0.8905 0.4473 -0.0501 -0.6741 -0.8071 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3722033
930 -1.0786 -0.8905 -0.2850 -0.0501 -0.6741 -0.8071 A124 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2643961
931 -1.5046 -0.8905 -0.5577 -0.7590 -0.6741 -0.8071 A41 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1381002
932 0.3139 0.6795 0.4434 1.3676 0.4307 -0.8071 A372 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.4496261
933 -0.7253 0.6795 0.0266 -0.0501 -0.6741 -0.8071 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.1874424
934 0.6602 0.6795 0.4434 1.3676 0.4307 -0.8071 A376 C147 1.3221312 0.9897072 0.3646278 1.3676416 -0.0080377 0.6973599 0.4990250
935 -1.2448 -0.0817 0.5136 -0.7590 0.4307 -0.8071 A206 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4161137
936 1.6024 -0.8905 2.2860 2.7854 0.4307 -0.8071 A419 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3527581
937 -0.2213 0.6795 0.4473 -0.0501 -0.6741 -0.8071 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2585864
938 -0.3079 -0.8905 0.4473 -0.0501 -0.6741 -0.8071 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3618200
939 -0.3789 0.6795 0.0266 -0.0501 -0.6741 -0.8071 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2451757
940 -0.4655 0.6795 0.0266 -0.0501 0.4307 -0.8071 A276 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2469113
941 0.1251 0.6795 2.2860 -0.0501 -0.6741 -0.8071 A319 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.6012197
942 1.4327 0.6795 2.2860 2.7854 0.4307 -0.8071 A421 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.2993663
943 -0.7253 0.6795 0.5136 -0.0501 0.4307 -0.8071 A272 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2018053
944 0.1563 0.6795 1.2342 -0.0501 0.4307 -0.8071 A303 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.4448777
945 -0.6300 -0.8905 0.4473 -0.0501 -0.6741 -0.8071 A186 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4078736
946 -0.7253 0.6795 -0.7836 -0.0501 -0.6741 -0.8071 A150 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3224757
947 -1.5912 -0.8905 -0.7914 -0.7590 -0.6741 -0.8071 A41 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1440541
948 0.6499 -0.8905 0.4473 -0.0501 -0.6741 -0.8071 A217 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.4307949
949 -0.5677 -0.8905 0.4473 -0.0501 0.4307 -0.8071 A227 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.4344258
950 -0.1260 0.6795 0.0266 -0.0501 0.4307 -0.8071 A276 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3034946
951 -0.0412 -0.8905 0.4473 -0.0501 -0.6741 -0.8071 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3173700
952 0.2983 -0.8905 1.1953 1.3676 0.4307 -0.8071 A358 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3546382
953 -1.5046 -0.8905 -0.7914 -0.7590 0.4307 -0.8071 A86 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2635390
954 0.4957 -0.8905 1.1953 1.3676 0.4307 -0.8071 A361 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3875382
955 -0.9920 0.6795 -0.2850 -0.0501 -0.6741 -0.8071 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2683704
956 0.5477 0.6795 1.2342 -0.0501 2.6404 -0.8071 A387 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.2289556
957 1.5972 0.6795 2.2860 2.7854 0.4307 -0.8071 A421 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3106607
958 -0.8032 0.6795 -0.7759 -0.7590 -0.6741 -0.8071 A150 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3656346
959 0.1493 -0.8905 1.1953 1.3676 0.4307 -0.4173 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3760979
960 0.3849 -0.8905 1.1953 1.3676 0.4307 -0.4173 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3709136
961 1.4292 0.6795 2.2860 2.7854 0.4307 -0.4173 A421 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.2349830
962 -1.1652 -0.8905 -0.7759 -0.7590 -0.6741 -0.4173 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1038294
963 -0.6543 -0.8905 0.4473 -0.0501 0.4307 -0.4173 A227 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4275606
964 -0.9504 0.6795 0.0461 -0.0501 -0.6741 -0.4173 A224 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2645643
965 0.2013 0.6795 1.2342 -0.0501 2.6404 -0.4173 A387 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.2361890
966 0.5633 0.6795 1.1953 1.3676 0.4307 -0.4173 A376 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3677182
967 -0.2144 -0.8905 0.4473 -0.0501 -0.6741 -0.4173 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.2812700
968 -1.1738 -0.8905 -0.7759 -0.7590 -0.6741 -0.4173 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1023961
969 -1.0006 -0.8905 -0.7759 -0.7590 -0.6741 -0.4173 A93 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1312628
970 -0.5590 0.6795 0.4473 -0.0501 -0.6741 -0.4173 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2606137
971 -0.5192 0.6795 0.4941 -0.0501 -0.6741 -0.4173 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2750470
972 -0.2750 0.6795 2.2860 -0.0501 0.4307 -0.4173 A339 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.5592589
973 -0.6231 -0.0817 0.4941 -0.0501 0.4307 -0.4173 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3565030
974 -1.0006 -0.0817 -0.7759 -0.7590 -0.6741 -0.4173 A116 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2333228
975 0.3035 -0.8905 1.1953 1.3676 0.4307 -0.4173 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3573469
976 0.0541 0.6795 -0.5577 -0.0501 -0.6741 -0.4173 A229 C326 0.4078813 0.4460352 -0.0758046 -0.2045827 -0.6741149 0.6597676 0.3834511
977 1.1694 -0.8905 2.2860 2.7854 0.4307 -0.4173 A416 C55 1.4810668 0.0181922 2.3175726 2.6287019 0.1225346 -0.2170130 0.3195137
978 -0.3616 0.6795 -0.5577 -0.7590 -0.6741 -0.4173 A146 C434 -0.4996726 0.3958051 -0.8430922 -0.8445203 -0.5645193 0.6623233 0.3303140
979 -0.5365 -0.0817 0.4941 -0.0501 0.4307 -0.4173 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3709364
980 -0.0862 0.6795 2.2860 -0.0501 -0.6741 -0.4173 A319 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.6382785
981 -1.3332 -0.8905 -0.7836 -0.7590 -0.6741 -0.4173 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1272090
982 0.3225 -0.0817 1.1953 1.3676 0.4307 -0.4173 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2257136
983 -1.0630 -0.8905 -0.7759 -0.7590 -0.6741 -0.4173 A93 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.1208628
984 0.8334 2.2969 1.2342 1.3676 0.4307 -0.4173 A403 C57 0.1798241 1.6748331 2.0467757 1.3131124 0.3882335 -1.9684654 0.6227230
985 -0.4811 -0.0817 0.4473 -0.0501 0.4307 -0.4173 A242 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.3723697
986 -1.0734 -0.8905 0.0422 -0.7590 -0.6741 -0.4173 A111 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2301755
987 -1.4180 -0.8905 0.0422 -0.7590 -0.6741 -0.4173 A111 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2511051
988 -0.1884 0.6795 0.0461 -0.0501 2.6404 -0.4173 A344 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.2254486
989 -1.0872 -0.8905 -0.7759 -0.7590 -0.6741 -0.5382 A66 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.0966794
990 -0.5677 -0.8905 0.4473 -0.0501 -0.6741 -0.5382 A198 C384 -0.0359246 -0.9572228 0.0505756 -0.1222502 -0.6547317 0.5368788 0.3603033
991 0.3554 -0.0817 1.1953 1.3676 0.4307 -0.5382 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2110469
992 0.1338 -0.0817 1.1953 1.3676 0.4307 -0.5382 A364 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.2237312
993 0.0627 -0.8905 1.1953 1.3676 0.4307 -0.5382 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3703812
994 -0.0394 -0.8905 1.1953 1.3676 0.4307 -0.5382 A351 C130 0.2826529 -0.0067137 1.6381330 0.9923522 0.2097590 -0.6177263 0.3873979
995 -0.4239 0.6795 0.4473 -0.0501 -0.6741 -0.5382 A246 C306 -0.7716661 0.5467115 0.2755179 -0.2228202 -0.1702066 -0.7871317 0.2629803
996 -1.5219 -0.8905 0.0422 -0.7590 -0.6741 -0.5382 A111 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.2482717
997 -0.5677 -0.8905 0.4473 -0.0501 0.4307 -0.5382 A227 C414 -0.4728276 -0.8354472 -0.4305269 -0.4820923 0.4307274 0.6016920 0.4332773
998 -1.0006 0.6795 -0.7759 -0.7590 -0.6741 -0.5382 A129 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3400395
999 -1.0214 -0.0817 0.0578 -0.7590 2.6404 -0.5382 A295 C142 0.1328959 0.6627507 0.4814828 -0.0248014 2.6404120 -0.9712527 0.5816155
1000 -1.0630 0.6795 -0.7759 -0.7590 -0.6741 -0.5382 A129 C478 -1.1829114 -0.7922801 -0.6999883 -0.7557764 -0.5234546 -0.6945646 0.3296395

hist(act_pred$diff, breaks = 30, col = "blue", main = "Mean Absolute Difference", xlab = "Difference")
Figure 22: Mean Absolute Difference

Figure 22: Mean Absolute Difference

7 Executive Summary

8 References

  1. Topology Preserving Maps : https://users.ics.aalto.fi/jhollmen/dippa/node9.html

  2. Vector Quantization : https://en.wikipedia.org/wiki/Vector_quantization

  3. K-means : https://en.wikipedia.org/wiki/K-means_clustering

  4. Sammon’s Projection : https://en.wikipedia.org/wiki/Sammon_mapping

  5. Voronoi Tessellations : https://en.wikipedia.org/wiki/Centroidal_Voronoi_tessellation